Re: Charles Lucy and Myself and the issue of "getting an established

> Posted by: "genewardsmith" genewardsmith@... ￜ genewardsmith

> LucyTuning is based on a fifth of 600+300/pi cents. Margo came up with a > tuning based on a fifth of 1200 (1+3e)/(2+5e) cents, where e = exp(1) is > the base of the natural logarithms. But Margo did not reject closely > related tunings as unacceptable, claim her tuning was the be-all and > end-all of musical tuning, or maintain her method even made sense other > than as a way which helped her break out the box and find a tuning she > liked and whose properties she could explore. She gained all the > benefits of eccentricity without being in the least eccentric. But if > making the tuning popular requires dubbing it something like > SchulterTuning and putting up a web site, patenting it, and endlessly > promoting it, maybe she fell short of the efforts of Charles Lucy. And > as I've remarked before, salesmanship does seem to have a good deal to > do with which tunings become well-known.

Dear Gene,

Thank you for a very accurate summary of my "e-based" tuning. And, of
course, we're absolutely agreed that it's only one possible shading in
a rich neighborhood: 46-EDO with a bit less tempering, or 63-EDO with a
bit more, for example. And while I happened to pick the point where the
ratio of the logarithmic (e.g. in cents) sizes of the whole tone and
diatonic semitone (at around 209.21 cents and 76.97 cents) is equal
to Euler's e, we could alternatively set that semitone to precisely
one third of a pure 8/7, for example, and get virtually the same tuning.

> I apologize in advance to Margo for dragging her into this, but it's the > only comparable example I know to LucyTuning.

Actually I'd like to thank you for your advertising <grin>, and add the
important point that while some tunings are promoted as "one size fits
everyone and all musical applications," the e-based tuning and lots
of others related to it are definitely _not_ in this category!

While you're well aware of what follows, Gene, I might include it as
a disclaimer to others:

If you're looking for thirds around 6/5 and 5/4, for example, you might
best look elsewhere. The focus is on regular thirds around 418 and 286
cents; neutral thirds around 341 and 363 cents (close to 15 and 16 commas
for followers of Near Eastern tuning discussions); and septimal thirds
around 264 and 440 cents. In my medieval and neomedieval styles, all these
thirds are pleasant but unstable intervals that resolve to stable ones
such as fifths.

Thus this is no replacement for meantone or other systems supporting
5-limit harmony: the outlook is quite different!

Anyway, Gene, I'm flattered that you would use this tuning as an example,
agree with the parallels and contrasts you've drawn, and would add that
at the time I devised it I was quite ignorant of Near Eastern music,
although now it's my favorite maqam tuning using a single 24-note
chain of fifths.

Best,

Margo

Dear Michael,

Please let me comment on your scale, and on David Keenan's
"Noble Mediant" which I was honored to be a part of in some
way, although it was definitely his idea that developed in
the process of some dialogues we were having in the late
summer of 2000 on tuning, and especially on complex intervals
such as thirds and sixths in a medieval or neomedieval style.

> Come to think of it...I've had similar issues with my (very old) PHI > sections theory as compared to high points of harmonic entropy. Because > if you take (using my old generating formula of (PHI-1)^x + 1 > exclusively...and no other tones):

Please just let me add early that I'd love to see your complete
scale, or maybe versions in different sizes.

> 0.618^1 + 1 = 1.618

Just to add a value in cents: this is, of course 833.090 cents, and the common element shared by the "Noble Mediant" approach and your scale.

> 0.618^2 + 1 = 1.3819 (near the area right between 4/3 and 7/5)

This is around 559.584 cents, a bit larger than 11/8 (551 cents) and smaller than 18/13 (563 cents). I wouldn't necessarily think of it as
maximum, just as an interesting place, although I haven't carefully considered this region. Curiously, your result does appear just about
identical to our Noble Mediant, but it's not something I would have
predicted (maybe someone like Gene could explain this result).

Something around this size could easily occur as a small "tritone"
in a maqam like Rast, and I'd happily use it in a polyphonic
progression, for example between the two upper voices of a sonority
with minor third (around 13/11 or 289 cents) and neutral sixth
(around 18/11 or 853 cents) expanding stepwise to fifth and octave.
This is a tad narrower, but should be fine! I'd use it first, and
ask questions about "harmonic entropy" later.

> 0.618^3 + 1 = 1.236 (not far from the mediant at (6+5)/(5+4) <the > mediant of 6/5 and 5/4> = 11/9)

Here I get 366.910 cents, which I'd consider in a different class than 11/9. although also, of course, a neutral third. I'd call it a submajor third, very close to 21/17 (366 cents) and not far from 26/21 (370 cents).
This would be a great third to use for a bright Maqam Rast, among
other things!

Anyway, our Noble Mediant is definitely different for 5/4 and 6/5, around
339.344 cents, or somewhere between 17/14 (336 cents) and 28/23 (340 cents). That's supraminor, a bit below the "central zone" roughly from
39/32 and 16/13 (342-360 cents or so), while yours is submajor.

So whatever is happening, you're not simply duplicating our results.

> 0.618^4 + 1 = 1.145 (about 8/7, not actually a mediant of any sort but > not at the 7/6 low entropy point on the harmonic entropy)

That's 235.774 cents, or a bit wide of 8/7. Our Noble Mediant is around
243.619 cents. Again, you're not duplicating our results but using a different method and getting different shadings of intervals.

Musically, the main distinction I'd see, and one noted by Jacques
Dudon in this general region also, is that while 236 cents might
still be like 8/7, a "large tone," around 240 cents we find that
an interval is large enough to serve as a very small "minor third,"
for example with one voice ascending by 60 cents or so and the
other descending by 180 cents or so, to contract to a unison.

For me, the exact degree of SHE (Cameron's Spectral Harmonic Entropy)
isn't as important -- if I could really determine it anyway -- as
the melodic context, and with music for two or more voices also the
vertical or contrapuntal context. How an interval helps form a
melody, or, if simultaneous, either marks a resolution (e.g. 3:2)
or helps move toward a resolution (as any of the intervals in your
tuning or almost any Noble Mediant might do), is most important.

Even with fine shadings, I don't necessarily think in terms of
exact "entropy" (as if such a thing were known), but in terms
of color -- with the realization, of course, that in many of
my styles 1:1, 2:1, 3:2, and 4:3 define stability (and 2:3:4
complete stability and richness), so those are the "valleys."
But there are important differences of color.

Thus around 355-360 cents is "high central neutral," while
365-370 cents is "submajor," a delightful small major third
at 21/17 or 26/21, for example! Both flavors are beautiful,
and it's nice to have both available. But to someone looking
for 5/4, both would be "high entropy" by comparison.

> 0.618^4 + 1 = 1.09 (about 12/11, again not actually a mediant of any > sort but not at the 7/6 'smallest ratio' entropy point on the harmonic > entropy)

Here I'm fairly confident you mean 0.618^5 + 1.

Indeed it's a near-just approximation of 12/11, 149.464 cents
by comparison to a just 150.637 cents. That's a central neutral
second, and in my view nowhere near 7/6, a small minor third;
those are different categories and regions of the spectrum.
But if I read you correctly, I do agree that 7/6 is maybe the
first dramatic "valley" as we move out from a unison and listen
to simple dyads. In many of my musical styles, it tends to be
the simplest and most concordant of all the thirds I regularly
use.

Of course any kind of second, and especially one substantially
smaller than 9/8, is going to be considerably more tense. But
in a 13th-century medieval European style, it's common to
use vertical minor seconds, so something like 1/1-3/2-18/11
with an upper 12/11 is by that standard "rather mild" <grin>.
Of course, the tension is something we (or at least I) welcome
in a sixth sonority seeking expansion to an octave. To give a
concrete example, in my present tuning I have A*-E*-F# (the
asterisk showing a note on the upper keyboard) at around
0-703-853 cents, resolving nicely to G*-D*-G*: the lower
two voices descend by usual tones at 207-209 cents, while the
highest voice ascends by a 138 cents or a near-just 13:12.

With 12:11 alone, the close spacing does add to the
tension. In two-voice counterpoint, say in maqam-based
polyphony, I might, all things being equal, consider
11:6 contracting to a fifth a bit milder than 12:11
expanding to a fourth -- but might well use either,
depending on the context.

Anyway, again, I would consider your 149-cent interval
as distinct from anything I'd expect from the Noble Mediant
method. And I tend to consider 12/11 as a cardinal point
of reference in itself, as it has been to various theorists
of maqam music looking at ratios at least since al-Farabi,
rather than a mediant of two other intervals.

> Some people saw a fair number of the fractions generated by ratios in my > "PHI sections" scale

Well, now I've seen a few, and would like to see more.

> A) Were near mediants between Harmonic Entropy

This seems to me a very general category. For example
either 339 cents (Noble Mediant) or your 367 cents is
in the neutral region, but they're near opposite ends,
respectively supraminor and submajor. This is a contrast
that I encounter and relish on an everyday basis, and
I can tell you that your result is different from what
Dave's method, which I was delighted to take some role
in discovering although it was his discovery, would
yield.

> B) They recalled that, historically, PHI had often been used for the > quest for maximum dissonance (even though I had been trying for maximum > consonance by using PHI in a different manner).

Phi can be used for many things, and one doesn't exclude another.

> So they threw their hands up and told me I had simply "discovered > someone else's theory". They said that the exact mediants were defined > by a method called noble mediants IE > http://dkeenan.com/Music/NobleMediant.txt that turn up numbers near > Harmonic Entropy maxima. But if you plug in the formula on that page to > Harmonic Entropy minima you will get ratios at least 10 cents away from > the values in my scale.

Yes, I agree that they are distinct methods, and arise from different
motivations. I'll explain the motivation that Dave and I had below.

> And even if the values were the same (or in fact, are close enough in > some people's minds to qualify as being the same), note that I generated > them by sections and Margo Schulter and David Keenan (authors of the > paper) generated them by mediants.

To me, both the methods and the results are distinct.

> Actually Margo, perhaps you could help explain the differences between > my old scale and the values from your "Noble Mediants" study > yourself...I figure it's clearest to get information from its original > source.

Well, while I can't speak for Dave, I see the results as very distinct,
in part because your scale touches a lot of the regions where I live
from day to day. The neutral third region from 330 to 372 cents or
so, say, is very richly populated, and 339 cents vs. 367 cents is
a distinction with a difference for me. In fact, those two intervals
together would give an approximation of 14:17:21 (0-336-702 cents), one
of my favorite divisions of the fifth into thirds!

> Now I step back years after I found the PHI-section scale. First of all, I
> disagree that it's as good at producing consonance as I originally thought. I
> originally thought, when rounded to JI values IE 8/7, 12/11, 16/13, 13/8,
> 11/8...it would be competitive with things like 1/4 comma meantone for
> consonance but enable more chords.

"Competitive" is to me a curious word, because these seem like two
different worlds to me. Here's a piece I wrote using some of the
ratios of the kind we're discussing, for example a near-just 13/8
and 7/4 as well a submajor third around 370 cents or 26/21, close
to your 367 cents:

<http://www.bestII.com/~mschulter/Prelude_in_Shur_for_Erv_Wilson.mp3>

This is from one point of view a variation on 13th-century European
polyphony using Persian Shur with its neutral steps and intervals
(or one possible flavor thereof). The same patterns apply from a
harmonic point of view: stable fifths and fourths, alternating
with various kinds of unstable sonorities from relatively blending
to impressively tense.

Now here's a short piece in meantone (actually 2/7-comma, and
a bit modified, but in the same general category as 1/4-comma):

<http://www.bestII.com/~mschulter/Invocation-ToneIV.mp3>

Major and minor thirds very close to 5/4 and 6/5 are stable
concords, playing a role analogous to 3/2 and 4/3 in the
previous piece. And there's less contrast in levels of
tension, as well as much greater restraint in introducing
unstable intervals (e.g. through suspensions).

In my view, neither style could substitute for the other:
and I love both!

And this brings me to my promised explanation of the Noble
Mediant. At the beginning of September 2000, Dave and I
were discussing medieval European music and its neomedieval
offshoots, and especially the role of complex major and
minor sixths at or near ratios such as 14/11 (418 cents),
13/11 (289 cents), 17/14 (336 cents) and 21/17 (366 cents).

His idea, when we discussed how some of these ratios are
classical mediants (e.g. 14/11 from 5/4 and 9/7 as
(5+9)/(4+7), was to find a way to locate the region of
maximum complexity between two simpler ratios.

On September 5, Keenan Pepper posted his proposal for
a "Noble Fifth" tuning based on a logarithmic use of
Phi -- not as an interval ratio, but as the logarithmic
ratio of sizes between a tuning's whole tone and
chromatic semitone -- in contrast to Kornerup's
famous Golden Meantone, with Phi as the logarithmic
ratio between the whole tone and diatonic semitone.

And Dave would want me to be _very_ clear on the
distinction between Phi as an interval ratio and
Phi as a logarithmic ratio!

Anyway, with our discussion on mediants and Keenan
Pepper's post as spurs, Dave got the idea that
Phi might be just what we needed to calculate a
weighted mediant showing the approximate region
of maximum complexity. And our paper followed.

As the paper itself explains, we were looking at
a situation where fifths and fourths are stable
concords, while major and minor thirds and sixths --
and sometimes neutral thirds and sixths also --
are used as unstable intervals.

While I consider the mathematical tool to be
wonderful, I should add that I'm not necessarily
going for maximum complexity, at least not all
the time, in such styles. Of course, for a region
like 408-440 cents or so, anything is going to
be relatively complex, and what I might seek is
some interesting shadings within this range,
which might but wouldn't necessarily have to
including something close to our maximum around
422-423 cents. Thus the Noble Mediant is a
great tool for orienting oneself and exploring
the different degrees and nuances of complexity.

> Now it doesn't seem as such for consonance but the range of chords > available with "at least decent" consonance still seems very broad to my > ears. Perhaps it's because the scale simply stays predictably in a range > of "decent consonance" and doesn't ping-pong from "great"/low to > "piercing"/high consonance (like so many microtonal scales do) that it > seems more stable than it actually is (at least in JI format).

A good starting point for your scale might be _timbre_. With
Bill Sethares, I'd say that a good tuning/timbre match for
you (and you're the judge, maybe with help from others) can
be far more important than someone's theoretical entropy
rating in the abstract. What kind of melodic, contrapuntal,
or harmonic style might fit your tuning/timbre combination --
or vice versa?

More focus on various traditional or new styles of melody
and polyphony -- the first not necessarily requiring the
second! -- and less theoretical micromanagement of someone's
notion of precise "harmonic entropy," might not be a bad
trend.

My advice would be that rather than trying to "compete"
with other scales, you should explore your Phi section
tuning as a world in itself. Maybe you'll want to develop
timbres that can maximize consonance by matching partials
or the like, especially if you want rather complex
sonorities to sound more independently concordant.

Remember, interesting melody is a fine starting point.
If you have that, then you can play around with matching
timbre to tuning, and get something really unique and
beautiful.

Try the Sethares "Xentonality" album, for example, for
a bit of orientation. Also, the Ethno2 collection and
pieces entered for the recent contest will give you a
sample of lots of world musical traditions. Can you
Phi scale approximate any of the tunings in that
collection, or suggest some interesting variation
on one or more those tuninings that you like?

Please forgive me for writing at this length.

Best,

Margo Schulter
mschulter@...

--- In tuning@yahoogroups.com, Margo Schulter <mschulter@...> wrote:

> > 0.618^2 + 1 = 1.3819 (near the area right between 4/3 and 7/5)
>
> This is around 559.584 cents, a bit larger than 11/8 (551 cents) and
> smaller than 18/13 (563 cents). I wouldn't necessarily think of it as
> maximum, just as an interesting place, although I haven't carefully
> considered this region. Curiously, your result does appear just about
> identical to our Noble Mediant, but it's not something I would have
> predicted (maybe someone like Gene could explain this result).

It's all due to the wonders of algebraic number theory; since 8 and 13 are Fibonacci numbers, 8+13 phi is a unit, meaning an algebraic integer whose inverse is an algebraic integer. Hence, the Noble Mediant is an algebraic integer, and we have:

1/phi^2 + 1 = 3 - phi
(11+18 phi)/(8+13 phi) = 3 - phi.

Gene>"1/phi^2 + 1 = 3 - phi
(11+18 phi)/(8+13 phi) = 3 - phi."

Funny because, you're right, that exact example does indeed yield exactly
the same answer. My question is where did/could you get 11/8 and 18/13 as the
inputs...other than reverse-engineering my formula (like you said...by finding a
ratio "a bit larger than 11/8 (551 cents) and smaller than 18/13 (563 cents)"
where 11/8 and 18/13 are the nearest fairly low-limit ratios (assuming 13-limit
or less) to the answer from 0.618^2 + 1 which = 1.3819?

Certainly it seems, in that case, the result IS a noble mediant between two
ratios, but not a mediant between any two "lowest entropy" ratios also on the
Harmonic Entropy curve. Margo, was not that the point of your paper, to capture
points of maximum entropy using the HE curve as a basis? It would seem to
follow that is you don't use low-HE ratios as inputs to the Noble Mediant
formula, you don't get mediants representing maximum Harmonic Entropy.

And, perhaps a side question, how would one go about finding a single
algorithm which would generate ONLY ratios of which taking the Noble Mediant
would produce the PHI sections scale?

>"Please just let me add early that I'd love to see your complete scale, or maybe
>versions in different sizes."

The full version, as I recall, is simply
A) The scale I just posted
B) PHI over the results I gave in part A......IE PHI/A (the "inverse" of the
scale AKA PHI / ((1/PHI)^x + 1))...I am pretty sure it's that and not "PHI minus
the results in A"...but I'll have to double check and post it in a bit.

Margo>"That's 235.774 cents, or a bit wide of 8/7. Our Noble Mediant is around
243.619 cents. Again, you're not duplicating our results but using a

different method and getting different shadings of intervals."

Even I will say...that one is very close, even if it is a different
"shading", close enough that I would not blame people for thinking they were the
same. What two fractions are you using as inputs to your formula to obtain
this?

>> 0.618^4 + 1 = 1.09 (about 12/11, again not actually a mediant of any
>> sort but not at the 7/6 'smallest ratio' entropy point on the harmonic
>> entropy)
>Here I'm fairly confident you mean 0.618^5 + 1.
Right, my mistake.

>"Anyway, again, I would consider your 149-cent interval as distinct from
>anything I'd expect from the Noble Mediant method."
Agreed.

>"My advice would be that rather than trying to "compete" with other scales, you
>should explore your Phi section tuning as a world in itself. Maybe you'll want
>to develop timbres that can maximize consonance by matching partials
or the like, especially if you want rather complex sonorities to sound more
independently concordant."

Coming back to this scale system years later...I indeed hear and feel how
much standard timbres are off...and at the same time how PHI^x-type timbres
won't work. Sethares' timbre to scale method is relatively well documented, but
his scale to timbre method far less so (admittedly it is likely beyond what I
know how to do mathematically...his tuning to timbre method is tricky enough to
analyze already). But, to make complex sonorities independently concordant or,
in my terms (as I understand it), maximize the number of tall chords available
that can work together regardless of if it does/doesn't acheive the overall
consonance of standard "non-PHI" theories, I agree that would be an ideal goal.

As for can my PHI sections scale approximate any tunings in the Sethares
Xenharmonic album collection...I don't remember seeing anything that matched
(lots of examples IE "Blue Dabo Girl" and "Ten Fingers" just make the timbre and
scale equal IE both aligned to 11TET and 10TET...and many others align to things
like a straight harmonic scale and timbre or a scale that matches a bell's odd
disharmonic timbre)...but I'll gladly check again.

--- In tuning@yahoogroups.com, Michael <djtrancendance@...> wrote:
> It would seem to
> follow that if you don't use low-HE ratios as inputs to the Noble Mediant
> formula, you don't get mediants representing maximum Harmonic Entropy.

That is correct. But it goes further. The inputs cannot be _any_ pair of low-HE ratios. They must be adjacent in some Farey series, or equivalently, adjacent on the Stern-Brocot tree (= Erv Wilson's Scale tree).

I suspect it's an equivalent requirement that if you take the product of each ratio's numerator with the other ratio's denominator the two products differ by exactly one.

For more explanation see this post and others in the same thread.
/tuning/topicId_73794.html#73809
Other useful stuff in this thread too
/tuning/topicId_76975.html#77376

There is a list of Nobles of various strengths here. Note that they do not admit of octave equivalence (or even "phi-tave" equivalence) in general.
/tuning/topicId_76975.html#77258

-- Dave

--- In tuning@yahoogroups.com, "dkeenanuqnetau" <d.keenan@...> wrote:

> That is correct. But it goes further. The inputs cannot be _any_ pair of low-HE ratios. They must be adjacent in some Farey series, or equivalently, adjacent on the Stern-Brocot tree (= Erv Wilson's Scale tree).
>
> I suspect it's an equivalent requirement that if you take the product of each ratio's numerator with the other ratio's denominator the two products differ by exactly one.

It is, assuming the ratios are reduced to lowest terms.

Here's a diagram showing the musically significant nobles in cents on a Stern-Brocot tree of the neighbouring ratios.
/tuning/topicId_77502.html#77502?source=1&var=0&l=1

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...>
> It is, assuming the ratios are reduced to lowest terms.
>

Thanks Gene.

"Musically significant" was a stupid thing for me to say. I should have said "Harmonically significant", in otherwords "audibly significant as an (approximate) frequency ratio" as opposed to say a ratio of step-sizes expressed as cents/cents. And of course it's only my opinion. Who knows what significance others may find.

--- In tuning@yahoogroups.com, "dkeenanuqnetau" <d.keenan@...> wrote:
>
> Here's a diagram showing the musically significant nobles in cents on a Stern-Brocot tree of the neighbouring ratios.
> /tuning/topicId_77502.html#77502?source=1&var=0&l=1
>