TUNING digest 1404

Bruce writes:
>Actually it seems to me that the fascination with polyphony along with
>12ET's somewhat universal simplicity, has caused the predominance of
>12TET-- it has taken 250-300 years to somewhat exhaust the rich endowment
>of this tuning.

If I understand Bruce to mean that ET has been used for 250-300 years, I
must respectfully disagree. There is more than ample evidence to show that,
while theorized about much earlier, the use of 12 TET didn't actually come to
prominence before 1850.
True, we have evidence that lutes and viols were capable of ET shortly after
the Mersenne ratios were published, (1636?), but Mersenne himself said that
keyboards could not use the numbers because they must be tuned by ear, not
linear measurements.
I hope we are not on a semantic loggerhead here, but the tunings that
evolved out of the restrictive meantone tunings were often called "equal" for
their modulatory freedom, but there was distinct differences between the level
of tempering in the various keys.
Regards,
Ed Foote
Precision Piano Works
Nashville, Tn.
http://www.airtime.co.uk/forte/history/edfoote.html

On Tue, 5 May 1998, Paul H. Erlich wrote:
> >(a) I wish to go beyond basic (level 1) consistency because it is
> >possible for an ET to be level 1 consistent and still err from a given
> >just ratio by nearly half the stepsize of the ET. An extreme example:
> >18TET is consistent to the 7-limit, but its 11-step "fifth" is only
> >barely better as an approximation to the 3/2 than its 10-step interval.
> >One is over 31 cents high, the other more than 35 cents low; the
> >difference between the two errors is less than four cents!
>
> That's a good reason, but it would seem highly unlikely that the most
> appropriate consistency level for this purpose would turn out to be an
> integer, unless you had particular composite ratios in mind that you
> care about representing consistently.

That's fair--you could, say, choose tunings where the roundoff error was
at most half the distance to the other alternative, which would result
in a cutoff consistency level of--uh--3/2, I think. Still, by and large
I think consistency levels make much more sense as integers, since they
correspond to secondary, tertiary, etc. intervals being represented
consistently. Why I care that they are, well, see below.

> It was the ratios of large
> composite numbers I was objecting to, since treating them more carefully
> than ratios of numbers with larger prime factors, even if the latter
> numbers are smaller than the former ones, is something that those who
> believe in prime characteristics will advocate, but I find no
> justification for.

Okay, so the question is, why would I care, for example if I was working
in 5-limit harmony, about how 9/8s or 10/9s are approximated, but not
any intervals involving 7?

Short answer: voice-leading.

Medium-length answer: I wish to approximate just intervals not merely in
isolated chords, but in the progression of one chord to another.
Representing the movement of one consonant identity to another without
producing (what are to me) bizarre perceptual artifacts requires level-2
consistency.

Long answer, with example: let's take this forum's favorite whipping
boy, 12TET. It's consistent to the 9-limit, so it ought to be okay for
some septimal harmony, yes? Let's take a 4:5:6:7 tetrad and move to
another 4:5:6:7 tetrad so that the 6 of the first tetrad becomes the 7
of the second. If we voice everything so that all voices move as little
as possible, the 7 of the first moves down by a 64/63, the 6 holds, the
5 moves up by a 36/35, and the 4 moves up by a 15/14. (Okay, so the
second tetrad is really in "first inversion" becoming 5:6:7:8.)

How is this represented in 12TET? A 4:5:6:7 tetrad has 4, 3, 3, and 2
steps between the members. If the first tetrad is rooted on 0 and the
second keeps the common tone, we have 0 4 7 10 moving to 1 4 7 9. IOW,
the 64/63 is represented by 1 step, the 36/35 by 0 steps, and the 15/14
by 1 step. To me, this doesn't make sense--since there's a common tone,
we should be able to hear the 8/7, 6/5, and two 7/6s relative to it
fairly accurately. Why should the 36/35 vanish when the much smaller
64/63 doesn't? But while I'm worrying about these rather large odd
numbers, ratios involving (say) 11 never entered into it.

I realize full well that these sorts of oddities are rich ground to
explore; indeed, a major part of Easley Blackwood's _12 Microtonal
Etudes_ springs from exploiting them. However, personally, my goals are
different.

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
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NOTE: dehyphenate node to remove spamblock. <*>

On Wed, 6 May 1998, it was written:
> Let's take a 4:5:6:7 tetrad and move to
> another 4:5:6:7 tetrad so that the 6 of the first tetrad becomes the 7
> of the second. If we voice everything so that all voices move as little
> as possible, the 7 of the first moves down by a 64/63, the 6 holds, the
^^^^^
> 5 moves up by a 36/35, and the 4 moves up by a 15/14. (Okay, so the
> second tetrad is really in "first inversion" becoming 5:6:7:8.)

I meant 49/48; please make the necessary substitutions. But the
principle remains the same.

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
-\-\-- o
NOTE: dehyphenate node to remove spamblock. <*>