RE: Decatonic tuning and 7-limit lattices

>}My point about tuning tritones isn't quite covered in Paul's paper.
>}To quote: "Since any perfect 6th10 in the scale is likely to
>}represent 7:5 and 10:7 equally often, the interval achieves its
>}greatest accuracy at exactly 600 cents." No, at least 1 of the 4
>}tritones in a pentachordal scale has only one interpretation. My
>}point is that, as at least two have to be ambiguous, these worst
>}intervals in 22-equal can't be improved by any tempering. A well
>}temperament may improve the average tuning, though, and I haven't
>}looked at this properly yet.
>
>You are correct, but with every new key one modulates to, my comments become
>more and more correct.
>
>}In abstract terms, the point of fcc is that, along with hcp, it
>}has the highest possible packing fraction for spheres.
>
>Correction: the highest _known_ packing fraction for spheres. The
>sphere-packing "theorem" is as yet unproved.
>
>}That's why
>}it's so common in crystals.
>
>Naah, crystal structures have little if nothing to do with the principle of
>close packing.
>
>}
>}
>} 5/3-------5/4
>} / \ / \
>} / \10/7 / \
>} O O / 7/6 \ / 7/4 \
>} | / \ / \
>} \___/ 4/3-------1/1-------3/2
>} \ / \ /
>} \ 8/7 / \12/7 /
>} \ / 7/5 \ /
>} \ / \ /
>} 8/5-------6/5
>
>}And there's probably a name for that structure, maybe 7-limit
>}tonality diamond.
>
>Yup, but it's the Wilson picture rather than the Partch picture of the
>7-limit tonality diamond.
>
>}In case people are getting deluged with all this theory, I'll
>}highlight the point that these lattices are very useful for
>}constructing 7-limit scales, and taking chords from them.
>
>I second that!


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From: alves@orion.ac.hmc.edu (Bill Alves)
Subject: Re: Da Capo Press e-mail
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Your experiments on pitch bend resolution are very interesting, but I would
like to make a suggestion. Determining pitch change by a "timbral shift" in
a rich harmonic spectrum seems to me to be a rather dicey method of
comparison, especially if you don't know what's causing it. How about
simultaneously playing a reference tone on another MIDI channel? That way
you can clearly hear a change in the beats if the pitch changes.

Bill

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
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From: Manuel Op de Coul
Subject: John Foulds
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Bill Alves made the following comment on my experiments:

> Your experiments on pitch bend resolution are very interesting, but I would
> like to make a suggestion. Determining pitch change by a "timbral shift" in
> a rich harmonic spectrum seems to me to be a rather dicey method of
> comparison, especially if you don't know what's causing it. How about
> simultaneously playing a reference tone on another MIDI channel? That way
> you can clearly hear a change in the beats if the pitch changes.

Thanks Bill for the excellent suggestion. I repeated some of the
experiments using that method with a sine wave voice, and agree it is
in fact a more reliable and sensitive technique. All pitch changes
that had been detected by the overtone method were confirmed as real
by the reference tone technique, but in addition some new pitch
differences were revealed that had escaped detection before. In
general the qualitative conclusions reached earlier were confirmed.

The reference tone method made it possible to tighten up the
quantitative estimates for the Sound Canvas (Roland SC-55mkII). It
appears that when set at a pitch bend range of +/-1, the Sound Canvas
has a pitch bend resolution of 8 bits everywhere except in the pitch
bend region from 8065 to 8319, where instead of the expected three
pitch changes there are none at all. The beat count shows that the
omission is not compensated for by a correspondingly larger pitch
change across that region. This would appear to be a design flaw. If
so it is an unfortunate one for microtonalists, because it introduces
a gratuitous additional error of 2.34 cents into intervals whose end
points are tuned by pitch bends of opposite sign (8192 being
equivalent to a pitch bend of zero). If it were not for this
aberration, Sound Canvas owners would enjoy a maximum error of
resolution of 0.78 cents. Another unit that I tested (Fatar GS 1000R)
did not exhibit this peculiar behavior.

Thanks also to John Loffink for his clarifying remarks, among
them:

> What you have uncovered is the underlying pitch resolution or pitch
> quantization of the instrument, whether adjusted by pitch bend or
> any other controller, envelope, LFO, etc. Manufacturers (except
> Symbolic Sound and Justonic) never publish these specs, and very few
> people have investigated them.

This being so we would do well to report to John anything new about
the resolution of particular models that we discover by
experimentation, so that he can (if willing) post the information on
his useful web site (http://freeweb.pdq.net/jloffink/default.htm).

--Bill Cooper, U.C. Berkeley


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From: Xou Oxno
Subject: Guitars in NYC: microtonal and otherwise...
PostedDate: 06-02-98 02:40:43
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