Re: Defining Meantone and Pythagorean; Erv Wilson

Dave Keenan wrote,

>>What's wrong with "chain-of-fifths".

If my understanding is correct, Erv Wilson used the term "linear
temperament" for the "chain-of-identical-fifths" idea. Is this designation
satisfactory for everyone?

Charlie Jordan wrote,

>It assumes that the repeated interval falls on the fifth note of the scale.
>In China, it is the fourth note.

Just to clarify, that is because Chinese music is based on a pentatonic, not
heptatonic, scale.

>In Joseph Yasser's 19-TET, it is the
>eighth.

Because his system was based on a 12-note scale. Strangely, this "eighth",
despite being tuned just like a 1/3-comma meantone fifth, was supposed to be
a dissonance in his system, as his "consonant" hexad consisted of the
"root", "third", "fifth", "seventh", "ninth", and "eleventh" in his 12-note
scale (a 2:1 would be a "thirteenth").

In my 22-TET, it is the "seventh", since my system is based on a decatonic
scale. In my paper, I call the approximation of 3:2 in any tuning system a Q
(for quint), and the term "chain of Qs" covers all these cases.

If you are familiar with Yasser but not me, I suggest you read my paper,
http://www-math.cudenver.edu/~jstarret/22ALL.pdf. Like Yasser, I look for
the next scale after pentatonic and diatonic, but I am guided by a Partchian
conception of harmony, though I maintain the concept that the basic
consonant chords should be formed by a fixed pattern of degrees in the
scale. 22-tET is not derived from numerology (as in Yasser's 7+12=19) but
happens to be the tuning in which such a scale can be found.

Dave Keenan wrote,

>Could be called the 22-tET region or the decatonic region or
>the Erlich region. Who was the earliest known proponent of 22-tET or
>tunings in general where 9 fifths is a major third?

Actually, if you want to call it the Erlich region, the major third is not
defined as 9 fifths but as a half-octave minus two fifths. Graham Breed
classifies 22 as diaschismic because 4 fourths is 2 major thirds (plus an
octave). Bosanquet may have been the first to discuss (in a not unfavorable
way) the relationship between Qs and Ts in 22-tET, but as far as I know, I
was the first to notice how economically the approximations to complete
7-limit harmonies come out in this system. I have searched the microtonal
literature for any mention of my decatonic scales but they seem to have been
missed. In fact, it is precisely as a result of that search that I became
acquainted with the literature. Erv Wilson favorably considered all kinds of
tunings, including some in the category in which 9 fifths is a major third.
It is interesting how on page 5 of http://www.anaphoria.com/xen3.html, he
considers scales with 9, 11, 12, and 13 nominal symbols. 10 is left out.
This is because it results not from a single chain of Qs, like the others,
but from two interleaved ones.

>If I were asked to determine the other limit of the Pythagorean region I'd
>put it at 17-tET, 705.8c (also 34-tET). This is where the optimum 4:5 flips
>from -8 to +9 fifths.

Of course, 34-tET has a much better 4:5 than those constructed by fifths. So
it doesn't really belong in this region, which I (and Graham Breed) would
call "schismic" rather than Pythagorean. 34-tET is diaschismic.

>So the known-world of chain-of-fifths tunings can be usefully divided into
>3 kingdoms:
> Chain-of-fifths
> / | \
> | Meantone | Pythagorean | Erlichean? |
> 691.5c 700.0c 705.8c 711.8c Borderline fifth size
> 59-tET* 12-tET 17-tET 59-tET Borderline ETs
>11 4 -8 9 14 Fifths in a maj third

17-tET and 59-tET are not even borderline in the Erlichean kingdom since
without the half-octave, you can't get the economical approximations I was
talking about. Perhaps it would be better off without my name.

>If my understanding is correct, Erv Wilson used the term "linear
>temperament" for the "chain-of-identical-fifths" idea. Is this designation
>satisfactory for everyone?

This is due to Bosanquet, and is the most natural and accurate term I can
think of.

Bosanquet also has terminology for most of the other stuff in this thread,
although not all of it is as good.

Graham Breed and I once had some discussion on how to define meantone.
While the argument for "any linear tuning in which +2 falls between 9/8 and
10/9" seemed natural, I believe we sort-of decided that meantone was not a
strict property of a tuning, but rather a way tunings are used -- some sort
of vague association between a +4 move and a 5:4 move (inspired by Wilson).

-C.

>>If my understanding is correct, Erv Wilson used the term "linear
>>temperament" for the "chain-of-identical-fifths" idea. Is this
designation >>satisfactory for everyone?

If so, it seems natural to then define Pythagorean as a linear temperament
where the generator is the perfect 3:2.

-C.

"Paul H. Erlich" wrote:.

>
> If my understanding is correct, Erv Wilson used the term "linear
> temperament" for the "chain-of-identical-fifths" idea. Is this designation
> satisfactory for everyone?

Paul!
I had to go to the horses mouth on this one myself for your statement has
an interesting slant to it!
Wilson state that a linear temperament will include "chain-of-identical-fifths"
but also any scale that can be interpreted as occurring on a chain of any
interval. He says that in order to show the full implications of what the
Bosanquet keyboard it would have required volumes that never would be
published. He sees these keyboard as a palette in which the composer is free to
fill in his own way. http://www.anaphoria.com/dal21.html on the bottom of fig
24 shows the template where the 3 is +1,9 +2, the 7 +10 and the 11 +18 in a
linear temperament of fifths. Since a "linear
temperament" of a "chain-of-identical-fifths" is included in his use of the
term you are using the term correctly and can still say what you want to say. I
expect that this might cause more questions that answers and will do my best to
answer, but please realize that Erv at this time cannot answer question through
me.
-- Kraig Grady
North American Embassy of Anaphoria Island
www.anaphoria.com