Yahoo Tuning Groups Ultimate Backup tuning "Rank-3" for newbies and oldsters -- link to article

Hello, all.

People here have joked about how no one yet understands "Rank-3"
temperaments; but in fact they can be, should be, and must be
explained in language that newbies -- and oldsters -- can understand.

This post is a link to a long article on this topic, together with an
executive summary so that people can see if they if they might be
interested in reading the full article. Special thinks to Chris
Vaisvil for the idea of "degrees of freedom," exactly what we need,
and the basis for the title of my article:

<http://www.bestII.com/~mschulter/tn101812-3degrees.txt>

Basically, if a "Rank-3" temperament means one with the three
parameters I'm about to describe, it's something a newbie or oldster
can understand without a lot of math. Let's try it!

Our temperament involves three "degrees of freedom," which I'll
describe in the simplest and most familiar terms I can. We need an
octave, which we'll set at 2/1; we need a fifth or fourth, which we'll
set at 704 cents or 496 cents. That's two of the three degrees of
freedom, and enough for us to tune a chain of fifths and fourths and
create a tuning, with 12 or 17 notes as a good size. Let's pick 12, in
order to keep things familiar.

Now comes our third degree of freedom: we're going to generate a
second 12-note chain just like the first, and place it at a _spacing_
that gives us some new types of intervals that we like.

So we have a choice of octave (here 2/1 or 1200 cents); generator
(here the 704-cent fifth or 496-cent fourth); and spacing between the
two chains we create using that octave and generator.

Very quickly, let's say we notice that we have regular major seconds
at 208 cents -- ten of them. We'd like a 7/6 third (267 cents), or
something close. That suggests a spacing of about 59 cents, to get us
ten pure 7/6 thirds.

We'd also like a 13/8 harmonic sixth, at 841 cents. We notice that we
have eight minor sixths at 784 cents -- a difference of 57 cents, and
another possible choice for our spacing, the third degree of freedom.

So we compromise and pick 58 cents, and tune that second chain. Now we
have a full-fledged "Rank-3" temperament -- at least if the term has
been intuitively defined <grin>: octave, 2/1 or 1200 cents; generator,
704 or 496 cents; spacing, 58 cents.

At this point, we know that we have lots of our beloved 7/4 harmonic
sevenths and 13/8 harmonic sixths within a cent or two of just; but
what happens next? What other intervals do we get? How can we use all
of this musically? To find out, just follow the link above.

Peace and love,

Margo Schulter

Chris wrote:

> Thank you for this Margo - this article is very informing - I
> appreciate it! And your deep knowledge of these tunings!

Dear Chris,

Thank you so warmly for your gracious words, for your idea about
"degrees of freedom" that launched my article, and also for your
understanding of how these tunings have been at the center of my
musical life for much of the time since the year 2000. And George
Secor was at it already in 1978, two decades before I started making
music with xenharmonic tunings!

Your generous words and your insightful allusions to physics provide
an opportunity, I hope, for a friendly dialogue on the relationship of
the Regular Mapping Paradigm (RMP) to the traditional or classic
methods many of us to discover and navigate tunings. This is an
opportunity for mutual understanding and discovery!

In physics, the powerful and often astounding tools and discoveries of
quantum mechanics haven't made classic Newtonian mechanics obsolete.
Bus and train schedules, and even schedules for travel in orbit,
generally rely on Newton's three familiar laws, and disregard the
effects of Einstein's special relativity which we know, nevertheless,
operate at walking or driving speeds as well as those near the speed
of light.

In music, I recognize RMP as obviously an intriguing and powerful
tool for creating new tunings, and also a way to get new perspectives
on tunings designed using other approaches, whether centuries ago or
in recent months and years by members of this list.

In fact, just yesterday I noted a 17-note tempered version I had made
of Manuel Op de Coul's pipedum_17c.scl using a system called MET-24,
rather like Peppermint but with a bit less tempering: 2/1 octave;
fifths at 703.711 cents; and spacing between 12-note chains at 57.422
cents.

The specific details aren't as important as the fact that I'd love to
get an RMP analysis of what commas I'm tempering out, and how the
accuracy would be rated using the same measures as for the tunings
designed using RMP methodology.

Clearly that would be a learning experience for me. And in return, I'd
be delighted to share what I know through lots of experience about the
region around 704 cents, and some of the landmarks of the region,
which recent tuning list discussions have caused me to scrutinize more
closely.

Of course, Chris, my invitation is meant for you and anyone else who
might be interested. Also, I'd be glad to help whoever is doing the
Wiki pages on 2-3-7-11-13 tunings update these pages to cover Geroge
Secor's 29-HTT of 1978 (which has a subset I feel absolutely deserves
recognition, like his Miracle tuning in a different area, as a
germinal landmark), Peppermint, O3, and MET-24.

That would, of course, involve not only my summarizing the data on
these tunings (with George Secor the best authority on HTT, of
course!), but someone involved with RMP figuring out the commas and
parameters and data for a listing. And in the process, I would get the
chance to learn more both about these tuning systems and RMP itself.

> Best wishes,
> Chris

With many thanks,

Margo

"Rank-3" for newbies and oldsters -- link to article

🔗Margo Schulter <mschulter@...>

🔗Chris Vaisvil <chrisvaisvil@...>

🔗Margo Schulter <mschulter@...>