Opinions

Dear friends,

Most of the messages I sent to tuning seemed to be considered as a second class information, a probable truth, mainly due to the lack of information I faced on many subjects which I didn´t have the opportunity to study in the university. These unfavorable fact get worse when wise members like C. Lumma, M. Battaglia, G. Smith, K. Pepper and other members entered to more complex areas which are far to be understandable to me. May be teaching courses by mail would solve this problem.

While I was satisfied with the future of the progression of musical cells and despite I explained its interesting properties, nobody gave a credit to this proposal. The age of the 12 tone scale has not ended, the multiple tone scales will not necesarily be the
solution. I don´t know if somebody has demonstraded whether not one of the twelve tone scale proposals complies with the listeners expectancy.

Along my stay in tuning I made some mistakes even in conexion with my
personal behavior, somebody should press the reset.

Thanks

Mario
November 04

Mario Pizarro wrote:

> Dear friends,

> Most of the messages I sent to tuning seemed to be considered as a
> second class information, a probable truth, mainly due to the lack
> of information I faced on many subjects which I didn't have the
> opportunity to study in the university. These unfavorable fact get
> worse when wise members like C. Lumma, M. Battaglia, G. Smith,
> K. Pepper and other members entered to more complex areas which are
> far to be understandable to me. May be teaching courses by mail
> would solve this problem.

Dear Mario,

Please let me report that I have looked into the Piagui tuning,
and cautiously conclude that it is a kind of well-temperament of
a kind much like that described by musicians such as Neidhardt
and Marpurg in the 18th century.

My purpose here is both to explain my tentative findings to you,
so that you may judge whether these findings fit your actual
scale; and also to explain them to others on the list, so that
they may understanding what Plagui is and what it does -- if I am
correct, that is!

Above all, this should be a friendly and mutually informative
dialogue. I suspect what has happened is that people have had
problems understanding some of your mathematical concepts leading
to Plagui, when the actual scale is quite simple, and certainly
within the legitimate scope of this group, just as similar
historical well-temperaments would be. My purpose is not to blame
anyone, but to suggest what Piagui may be, and seek your
feedback.

In my analysis, there are two sizes of semitone in Plagui. The
larger semitone or P is equal to precisely half of a 9/8 tone, or
101.955 cents. This is the meansemitone of 9/8, as it might be
called, famously favored by Henricus Grammateus in 1518, who
showed a geometric method for calculating a precisely equal
division of 9/8 in "an amusing reckoning," as he called it.

The smaller semitone or K is equal to (128/81)^(1/8), or
precisely an eighth of a Pythagorean minor sixth at 792.180
cents, or 99.0225 cents.

The Piagui scale, as I understand, has 8 K plus 4 P to form an
octave of 2/1. This makes sense, since the 4 P will add up to a
Pythagorean major third at 81/64 (407.820 cents), and the 8 K, of
course, to 128/81.

These 8 K and 4 P might be ordered in various ways, but a version
I saw on a webpage of Chris Vaisvil seems musically logical:
there the pattern is K K K K P P K K K K P P. This has the effect
of yielding a just 4/3 and 16/9 above our 1/1, here taken as C in
the well-tempered circle; and also of yielding above this 1/1 the
best major third from a classical perspective favoring 5/4:
396.090 cents, or 9.776 cents wide.

<http://chrisvaisvil.com/?p=564>

! piaguilike2.scl
!
Like Mario Pizarro's Piagui: steps of (9/8)^1/2 and (128/81)^1/8
12
!
99.02250
198.04500
297.06750
396.09000
4/3
600.00000
699.02250
798.04500
897.06750
16/9
1098.04500
2/1

There are two sizes of fifths, eight at 1/8-Pythagorean comma
narrow of just (699.0225 cents); and four at a just 3/2
(701.955).

We can also write out this temperament as a circle of fifths,
starting for example from Eb, with 4 fifths at a just 3/2 and the
other 8 at 1/8 Pythagorean comma narrow:

-1/8 3/2 3/2 -1/8 -1/8 -1/8 -1/8 3/2 3/2 -1/8 -1/8 -1/8
Eb Bb F C G D A E B F# C# G# D#/Eb
297 996 498 0 699 198 897 396 1098 600 99 798 297

Also, I might ask if your reference to "624" could be to (9/8)^6,
or a 2/1 octave plus a Pythagorean comma at 531441/524288; and
your "612" to 612 steps to the octave? While these concepts may
have played a role in leading you to Piagui, the system itself
seems a mild well-temperament of a general type well-known to the
18th century.

This style of temperament might also have some resemblance to the
"Victorian" tunings of the later 19th century, which sought
approximately equal semitones, but with some favoring of the
nearest keys. The K K K K P P K K K K P P generally has this
effect for 5/4, and so could be seen as another variation on
either the earlier Neidhardt and Marpurg temperaments in this
general style, or the Victorian tradition of subtly unequal
semitones and key colors.

> While I was satisfied with the future of the progression of
> musical cells and despite I explained its interesting
> properties, nobody gave a credit to this proposal. The age of
> the 12 tone scale has not ended, the multiple tone scales will
> not necesarily be the solution. I don't know if somebody has
> demonstraded whether not one of the twelve tone scale proposals
> complies with the listeners expectancy.

Mario, I am delighted to give you credit for the Piagui
well-temperament, and to recognize it as one of a number of
moderate well-temperaments quite close to, but distinct from,
12-EDO, which is how you often describe it.

Of course, it is possible that on further searching we might find
something identical to it from some 18th-century source, say --
and that would make it an "independent rediscovery," something
yuu devised in your own right, and later found had been described
earlier. This might be true of almost any tuning: but in any
event, I would call it a mild and subtle well-temperament with
four pure 3/2 fifths and eight tempered at 1/8 Pythagorean comma.

> Along my stay in tuning I made some mistakes even in conexion
> with my personal behavior, somebody should press the reset.

This is actually a good opportunity for all of us to "press the
reset" by recognizing your tuning as a mild well-temperament, and
approaching it on that basis.

One thing i would urge you to do, early in your presentations of
Piagui, is to give your Piagui scale in cents, and also the sizes
of the K and P semitones in cents. That way, people can quickly
understand what the actual tuning is. I think that a number of us
had problems understanding the basics of the scale, and this
complicated things a bit.

Also, I'm not sure how Ramos ties in with Piagui, at least in any
direct way: Henricus Grammateus, Neidhardt, and Marpurg all seem
much more immediately relevant. And we could go back to
Philolaus, as recounted by Boethius, who speculated about
dividing a Pythagorean comma into two equal parts, so that a
literal half-tone or (9/8)^(1/2) would be possible. He correctly
concluded, according to Boethius, that this half-tone would be
equal to a limma or diatonic semitone at 256/243 (90.225 cents),
plus the half-comma (11.730 cents).

In 1482, Ramos described a just intonation monochord based on
ratios of primes 2, 3, and 5; and a 12-note keyboard very likely
in some form of meantone temperament. He does not, to the best of
my knowledge, discuss any form of equal temperament. Nor does
give any mathematical details on his practical keyboard, although
Mark Lindley has shown why meantone temperament is the logical
conclusion based on the "good" and "bad" thirds and other
intervals which Ramos describes.

What immediately helped me form my hypothesis as to how Piagui is
tuned was the Scala file at Chris Vaisvil's site. I could look at
the cents, figure the sizes of the semitones P and K, and then
reasons as to their likely mathematical definition.

If you discuss Piagui as a well-temperament, then people will be
able either to understand immediately what it is, or at least to
look up well-temperaments and fit yours into the history of
tunings while understanding what some of the advantages and
disadvantages might be.

Peace and love,

Margo Schulter