Yahoo Tuning Groups Ultimate Backup tuning 17-WT paper in XH18 & comments about 34 and 46-ET

This is in reply to Gene's message:
/makemicromusic/topicId_14872.html#14946
regarding my 17-tone well-temperament paper that was released in
Xenharmonikon 18. This is now available online via a link in this
message:
/makemicromusic/topicId_14872.html#14943
in which there's also a link to a 17-WT jazz example (mp3 file)
mentioned in the paper. Comments are welcome.

--- In MakeMicroMusic@yahoogroups.com, "Gene Ward Smith"
<genewardsmith@...> wrote:
>
> --- In MakeMicroMusic@yahoogroups.com, "George D. Secor"
> <gdsecor@> wrote:
>
> > Hurrah! The 17-tone paper (from XH18) is finally online:
> > http://xenharmony.wikispaces.com/space/showimage/17puzzle.pdf
>
> To some small extent, this paper reads like a brief in favor of 34
> tones.

My thinking about 34 has recently changed quite a bit (in favor of
it), so I'm much more inclined to agree with you now than when I
wrote the paper (almost 5 years ago).

> You feel that diatonic melody is superior with fifths sharp
> around this size, but also that diatonic harmony is nonexistent.

I'd prefer to substitute "traditional harmony" for "diatonic harmony"
in the above sentence. There is a diatonic scale in 17, but it's
superpythagorean, or non-5, in nature.

> The
> most direct answer is Pajara/Diaschismic[14], 14 tones of two
> parallel chains of fifths. Or if you can wean yourself away from
> those chains of fifths, Pajara[10] or Pajara[12] a la Paul Erlich.

Yes, it's nice that you mention this. Just last week I wrote Graham
Breed off-list with some comments about his excellent article, _The
Regular Mapping Paradigm_:
http://x31eq.com/paradigm.html

One portion of my comments, which I'll quote here, concerns a
statement about applying the criterion of consistency when evaluating
ET's:

BEGIN COMMENTS:

I take issue with this statement:

<< Once you can brand them inconsistent you can also consider them
defective and so ignore them. >>

I find it tragic if someone interested in the 17 limit took this
advice and thereby ignored both 34 and 46-ET. It is only on account
of a single interval (13:15) and its inversion that 46 is 15-limit
inconsistent [might I add: 17-limit inconsistent], and the
inconsistency is so slight as to be inconsequential.

In the case of 34, should we take 28deg34 rather than 27deg34 for
4:7, then the inconsistency completely disappears! While it's true
that we've increased the amount of error for 4:7 (to +19.4c instead
of -15.9c), it should be taken into account that (as with major 3rds)
a positive error for the harmonic 7th is more acceptable melodically
than a negative error of similar magnitude; try it and "see" for
yourself. This is also consistent with using 17-ET (subset of 34) as
a 13-limit non-5 tuning. This is also the way prime 7 is mapped in
pajara (which is arguably better in 34 than 22, if you weight the
error of the intervals according to their relative consonance; again,
listen for yourself). Besides 4:7, there is only one other 17-limit
consonance that would not have its best value: 14:17 (and its
inversion). Furthermore, I found that if I put two circles of my 17-
tone well-temperament (see Xenharmonikon 18) 600 cents apart, then
the best 4:7 in the best keys is in fact 28deg34, thereby making
inconsistency a non-issue.

With all the interest in 17 lately, I suspect that I'll be writing a
paper about 34 one of these days.

[And from a subsequent message, re 22 vs. 34]
... Even though 22 is consistent at the 11 limit, you'll observe that
the 6:7:9:11 chord doesn't come off well at all (on account of a bad
9:11), whereas 34 is much better. As you already know, at the 5
limit it's no contest.

END OF COMMENTS

Let me draw attention to my statement that 34 is probably better than
22 for pajara; I would say that the improvement is comparable to that
of 31 vs. 19 for meantone. I'm thinking of melodic as well as
harmonic considerations, particularly in that both 19 and 22 have
narrow major 3rds, which are not so desirable.

> Both the harmonic and melodic resources are rich enough to began
> with, and of course if you've gone ahead and actually tuned an
> instrument to 34 notes, as some people have managed sucessfully,
you
> have complete harmonic freedom--no well temperaments need apply.

I don't think I would completely agree with that last phrase. With
34-WT (taken as two circles of 17-WT, as mentioned above), there
would still be complete harmonic freedom, plus the additional
dimension of differences in intonation (or "mood") in different keys.

OTOH, the well-tempering improves only primes >5, so it's arguable
that WTing is really not beneficial in 34.

I would need more time to decide.

> I note also you mention 41 but not 46 in your paper; 41 is a
> schismatic system, but 46 has fifths in the range you regard as
> melodically ideal.

Yes. Another problem I find with 41 is with prime 5: not only is the
tempering rather excessive for so many tones, but the major 3rd is
narrow (and therefore less desirable melodically).

> Of course, for many purposes both have too darn
> many notes.

Yes; in the mid-70's I was of the opinion that 31-ET (and, by
implication, 34-ET) was around the upper limit of what could be
considered practical for woodwind and valved brass instruments.

> Have you or Margo ever toyed with 46? You've mentioned you like it
> better than 31, and it seems pretty well up your and Margo's line.
> Smallish leading tones of 78 and 52 cents are of course available.
> The leapday temperament, using 27 degrees of 46 (the fifth) as a
> generator, has MOS of 12, 17 and 29 steps. The Pythagorean thirds
are
> much, much closer--nearly pure--to 14/11 than in the case of 17.

Yes, I've played with it in Scala from time to time.

> The
> 17 MOS therefore has some of the advantages of a well-temperament,

Not so -- the best keys of 17-WT have *wider* 5ths than 17-ET, which
improves primes 7, 11, and 13.

> whereas the 29 MOS has a variety of harmony including the good 5-
> limit harmony of 46, but with all kinds of exotic possibilities. Of
> course if you get away from fifths as generators other
possibilities
> open up, some involving rather small scales.

I haven't looked at 46 as closely as you have, but I agree that it
has a lot of possibilities.

--George

17-WT paper in XH18 & comments about 34 and 46-ET

🔗George D. Secor <gdsecor@yahoo.com>

🔗Gene Ward Smith <genewardsmith@coolgoose.com>