A pitch for making synthesizers tunable to an accuracy of at least 0.1 cent

>I believe that the difference between being able to set a
>pitch to within plus or minus 1 cent and being able to set that pitch to
>within plus or minus 0.1 cent could in some cases spell the difference between
>achieving a really powerful musical effect

I'm inclined to give that the ol' "yes and no" response. I agree very
definitely that accurate rendering of pitch can present some intriguing
differences in how harmonies are perceived. Thus JI vs. 12TET vs. various
other temperaments.

But I think it's also worth pointing out that, most real-world
instruments (keyboards being a very important exception) are a very
different story. The vast majority of instruments are imprecise-pitch
instruments, meaning that their players have to constantly monitor their
pitch, often correcting flaws in the acoustic design of the instrument
itself.

But the reality of the matter is that for most instruments, it's often
difficult to play normal-speed music with 5- to 10-cent accuracy much less
1-cent accuracy. Not impossible, true, but difficult. I often practice my
tenor saxophone with the tuning meter turned on constantly, and I'll look
down to see notes sometimes over a comma off the nominal 12TET pitch.
Granted, that error is probably what causes me to look down at the tuner.

Or to present this in a more extreme light, I'm currently struggling
with some sort of really obnoxious tuning problem on my soprano saxophone.
Certain pitch ranges on the horn routinely come out (without correcting
them) nearly a quartertone off! It's extremely difficult to correct the
pitch of an instrument that much, so even after correcting them, those
notes almost always end up much more than a comma off. And this is
otherwise a high-quality instrument (Yamaha YSS-675, retailing for well
over $3000), and at least one historically very good repairman couldn't
find anything wrong with it!

So my point is this: I agree that there are intriguing subtleties for
sustained chords down in the 1-cent-resolution range. But it's very
important to understand that people in general are used to VASTLY more
"slop" in pitch than that. They don't routinely operate down in the
pitch-discrimination realm of single cents for normal-speed music. I
therefore personally find more intrigue in tunings that produce entirely
different intervals (e.g., 9:7 instead of 5:4), or entirely different scale
structures (e.g., no meaningful approximation to a diatonic scale).

To me personally, entirely new harmonic resources and melodic scale
structures are more interesting than one- to five-cent distinctions of the
temperament of a well-known scale structure.

On Sun, 14 Jun 1998, Paul H. Erlich wrote:
> Barbershop harmony seems to be able to take the septimal comma, which
> would be 27 cents in just intonation, and by adjusting the (harmonic
> and/or melodic) intervals in the comma's context, and/or shrinking the
> comma itself, make it an effective "performance comma".

I'm curious about this. I assume Paul E. is talking about the
reinterpretation of the traditional dominant seventh chord as a 4:5:6:7
tetrad. However, this would imply an interval of 16/9 between the root
and the seventh, like so:

5/4

16/9 [4/3] 1/1 3/2

and I'm not sure why one would choose 16/9 when that makes that pitch
totally unrelated (within the 5-limit) to any of the other pitches in
the chord. Well, I mean, obviously it's the distance between the 4/3
and the 3/2 (3/4) of the key, critical pitches if you're using the dom7
actually as a _dominant_ 7th. But considered purely as an isolated
sonority, within the 5-limit I would tune it thus:

5/4

1/1 3/2

9/5

(Arrgh! This is why I prefer ETs. In an ET in which the 81/80
vanishes, these two are the same and you don't have to worry about it.)

I'm not very familiar with the barbershop style. Do they generally use
the dominant 7th as V7 of the key, as opposed to jazz harmony which uses
it as a sound color that can occur almost anywhere, functionally?

The point is, given this interpretation the distance to the 4:5:6:7 is
36/35, which is the septimal comma plus the syntonic comma or nearly 49
cents, a significantly larger interval to swallow as a comma--though
still not as big as the 648/625 I mentioned in a previous message.

> >Now, the reason I started looking at this is because I have been
> >trying to work out some ideas about consistency. It occurred to
> >me, as both Pauls reactions seem to confirm, that the ancient notion
> >of a 'comma' is (kind of) the original root of the idea of
> >consistency.
>
> I don't see it.

I'm not sure, but I _may_ see it--if we require that a comma, meaning
any just interval that vanishes in a particular ET, be smaller (in its
just version) than any of the non-vanishing intervals in that ET--which
seems reasonable to me--that has certain implications as to the
approximation errors of the primary intervals in relation to stepsize,
and hence to consistency. It's not a _direct_ relationship (complexity,
or "n-ary-ness" as Patrick and I have been calling it in private
correspondence, comes into it as well, plus some other stuff too), but
it is a close one.

> >(i) choose the n-ET and the m-limit to be considered;
> >(ii) list the intervals which belong to the primary m-limit;
> >(iii) find the primary m-limit interval which is least well
> >approximated by the chosen n- ET;
> >(iv) add this interval to itself until the deviation from its
> >intended equivalent is greater than (1200/2n) cents; call the number
> >of additions at this point t.
> >(v) then n-ET is level-(t-1) consistent at the m-limit.
>
> This certainly doesn't work for level 1 consistency. Is this the thing
> that happens to work for higher levels? If so, how are you defining
> "primary" and "number of additions"?

We went over this when I first posted my algorithm to the list--the
difficulty comes in how the roundoff error of, say, the 5/3 is
calculated. Rather than calculate the size of the 5/3 in steps of the
ET and finding the roundoff error directly, one must calculate the
roundoff error of the 5/4 and the 3/2 separately, and then subtract
them. Given that, the algorithm works for level 1 as well as higher
levels of consistency. It was easier to see from the pseudocode I
posted months back.

--pH http://library.wustl.edu/~manynote
O
/\ "Churchill? Can he run a hundred balls?"
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Paul H.,

Commas typically arise not within individual chords but over some span
in a piece of music, where various harmonies are constructed from a
given pitch-class set. The septimal comma in barbershop harmony -- the
typical progression is several dominant sevenths, each the dominant of
the next. If the roots are tuned Pythagorean, and the chords 4:5:6:7,
then the septimal comma arises between the seventh of one chord and the
root of the chord two chords later, which notes are the same pitch
class. In an even more typical IV-V7-I progression, the septimal comma
arises between the root of the first chord and the seventh of the second
chord. In such an instance, "hiding" the comma is particularly important
for maintaining the coherence of the pitch-class set, in this case the
major scale.

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