Re: 22-tET

on 2/4/01 8:15 PM, tuning@yahoogroups.com at tuning@yahoogroups.com wrote:

> From: "Paul H. Erlich" <PERLICH@ACADIAN-ASSET.COM>
> Subject: RE: Re: kabbalah

>> When was your paper posted on tuning list, let me know, and I'll search the
>> archives to read it.
>
> It wasn't. Go to http://www-math.cudenver.edu/~jstarret/22ALL.pdf (but
> ignore page 20 -- it's hopelessly erroneous).

Thanks Paul, will do.

> I don't know what your 31-tET beast is, but no, you can't get at any of the
> interesting properties of 22-tET by using a subset of 31-tET. They're
> completely different animals.

My 31tET 'beast' is a large fretted dulcimer/zither like creature. In the
same way that I've been able to approximate various 5, 7 and 11 limit tones
with it, I was hopeful that there might be some overlap to at least check it
out. C'est la vie.

Seth

--
Seth Austen

http://www.sethausten.com
emails: seth@sethausten.com
klezmusic@earthlink.net

Seth Austen wrote,

>In the
>same way that I've been able to approximate various 5, 7 and 11 limit tones
>with it, I was hopeful that there might be some overlap to at least check
it
>out. C'est la vie.

Though 22-tET, like 31-tET, is consistent through the 11-limit, its logic is
totally different. 31-tET, being a meantone tuning, follows the triadic
diatonic logic that Western music has used for the last 500 years, while
22-tET doesn't. Meanwhile, 31-tET can't support the decatonic scales that I
describe in my paper on 22-tET. Both tunings will be able to approximate
most of the JI chords you're probably interested in, while each tuning
offers some wonderfully consonant chords not available in JI or in the other
tuning.

Paul Erlich wrote:

< Though 22-tET, like 31-tET, is consistent through the 11-limit, its logic
is
totally different. 31-tET, being a meantone tuning, follows the triadic
diatonic logic that Western music has used for the last 500 years, while
22-tET doesn't. Meanwhile, 31-tET can't support the decatonic scales that I
describe in my paper on 22-tET. Both tunings will be able to approximate
most of the JI chords you're probably interested in, while each tuning
offers some wonderfully consonant chords not available in JI or in the
other
tuning.>

What are the consonant chords available in 31tET and 22tET that are not
available in JI ?

Surely any irrational no can be satisfactorily approximated by RI and if
the chords are consonant then they must therefore approximate JI chords ?

Justin White

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Justin wrote,

>What are the consonant chords available in 31tET and 22tET that are not
>available in JI ?

>Surely any irrational no can be satisfactorily approximated by RI and if
>the chords are consonant then they must therefore approximate JI chords ?

Ah, well I was using the Dave Keenan definition of JI here. Sidestepping the
definitional issue, let's call these "necessarily tempered" chords.

Consider chords produced by a chain of fifths or fourths, for which the JI
version would of course be Pythagorean tuning.

In 31-tET, a stack of fifths will contain some intervals that are more
consonant than those in a Pythagorean stack of fifths -- for example, four
fifths produce a 5:1 almost exactly. I would say that a chord of 4-7 notes
in a stack of fifths, with some octave reductions possible, is more
consonant in 31-tET than in Pythagorean tuning.

In 22-tET, a stack of "fourths" (approximate 4:3s) will contain some
intervals that are more consonant than those in a Pythagorean stack of
fourths -- for example, three "fourths" produce a very good 7:3. I would say
that a chord of 4 notes in a stack of "fourths", with some octave reductions
possible, is more consonant in 22-tET than in Pythagorean tuning.

In order to get those ratios of 5 or 7 to come out in JI, you'd have to
temper one of the fifths or fourths by a syntonic or septimal comma. But
that would lead to a nastily beating fifth which surely isn't considered
consonant.