Blackwood's proof

Mark Nowitzky writes:

> It was something about Helmholtz'
> and Ellis' attempt to use just intonation to get better harmony than equal
> temperament being proved impossible by Easley Blackwood in "The Structure of
> Recognizable Diatonic Tunings".

I don't know how familiar folks are with Blackwood's book. I'm not any
kind of musician. But I really loved the book, because it built a bridge
between my own thinking and more conventional musical ideas.

Before I read Blackwood I had tried taking a couple of Bach harmonized
chorales and figuring out how they should be played in j.t., I think 5-limit
is the right term. Basically the problem is just to figure out which of a set
of nearby pitches to use for each note, the pitches differing by the 80/81
comma. Being a musical ignoramus I'm sure I made tons of errors. But the
exercise really opened up for me the structure of the Bach pieces, or
seemed to anyway. I found, though, that the pieces seemed to turn on a kind
of musical pun, to take advantage of the blurring of harmonic distinctions
provided by temperament.

As I read Blackwood, the point is not so much that e.t. or any temperament is
so much better than j.t. in any universal or absolute sense, but that
mainstream European musical practise at least in the 18th & 19th Centuries
was really structured around temperament, and you can't just slide j.t. into
the classics and make them better. A different kind of music based on
j.t. might be better in some sense, but the structure of composition is
strongly connected to tuning, so you can't change one without some adjustment
in the other.

Jim



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From: clumma@nni.com
Subject: Barbershop and Equal-Step Tuning
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On Thu, 25 Sep 1997, Carl Lumma wrote:
> >Peter, if you are interested I could dredge up some of the messages from
> >the last time we discussed this. (Hey, is the list archived anywhere?)
>
> I for one would be interested in anything you've got on this. I think I
> remember seeing this term "consistency" in an article in an old issue of the
> Xenharmonikon, but I didn't understand it until now. Thanks, Paul!

I'll start culling through my files to find the messages that are worth
reposting.

> There was also another term that Ivor used, "symmetry". He never defined it
> too well, but he complained that 12-tone was too symmetric, and that when
> one first began Xenharmonic exploration, it was crucial to "knock the ear
> out" with a non-symmetric tuning like 19-tone. Could this be the same
> thing, Ivor's term for consistency?

I don't think so . . .

> Wouldn't it be nice to have a chart that measured both the "approximation"
> and "consistency" values for each limit for each temperament?

Look at these URLs:

http://library.wustl.edu/~manynote/consist.txt
http://library.wustl.edu/~manynote/consist2.txt
http://library.wustl.edu/~manynote/consist3.txt

The first shows what Paul E. and I were calling "consistency level" for
all ETs up to 1200TET. A tuning is level N consistent at a certain
harmonic limit M if not only all the M-limit intervals are consistently
represented, but also up to N combinations of them. Example: because
12TET is level 3 consistent at the 5-limit, we know that three 6/5s (3
steps), or 216/125, will be most closely approximated by 9 steps.

The second shows the same information as the first, but only displays an
ET if there is some limit at which it has a higher level of consistency
than all lower-numbered ETs, going up to 10000.

The third shows the greatest error in cents for each harmonic limit an
ET is consistent within, up to 99TET.

--pH http://library.wustl.edu/~manynote <*>
O
/\ "Well, so far, every time I break he runs out.
-\-\-- o But he's gotta slip up sometime . . . "



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From: Paul-Hahn@library.wustl.edu
Subject: Symmetry (was Re: Barbershop and Equal-Step Tuning)
PostedDate: 25-09-97 22:13:30
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There is also another table like Paul Hahn's with the consistency limits
not only of integer octave divisions but of every equal temperament up to
650 tones per octave. At:
ftp://ella.mills.edu/ccm/tuning/papers/consist_limits.txt

A larger file ftp://ella.mills.edu/ccm/tuning/papers/cons_limit_bounds.txt
contains the step size bounds for each consistency limit of 3 to 18,
also up to 650-tET.

Manuel Op de Coul coul@ezh.nl



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From: PErlich@Acadian-Asset.com
Subject: Re: 22tet and 24tet
PostedDate: 26-09-97 20:28:13
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