P. Erlich's comments

> Well, you define the "best" syntonic comma as the one which produces the
> best major third. The implication of notating a minor third as a pythagorean
> minor third plus the "best" syntonic comma is that it should be the best
> minor third, i.e., closest to just. When a better minor third is available
> in the tuning, which happens in 20tET, and even 17tET which you describe on
> the fifth page of your paper, the notation will obscure this fact. So I
> object to your definition of a "best" comma.

"Best" has a strict mathematical, as opposed to musical definition.
Consequently, the most desirable musical result may conflict. Of course
you may rework the notation to reflect that. I chose not to, simply
because I was aiming at a general method. The method accounts for all the
usual notations of many ETs, such as 19, 22, 31, 43, 53, 72, and others,
including many not usually used. It also allows for an improvement in
Blackwood's notations in a few cases.

I disagree with Paul E. that there is an implication that the best minor
third is what he says. Obviously it will frequently emerge from the
complement of the M3 in the P5, but certainly in 20 it doesn't.

However, it does seem possible to build a notation out of various
intervals; using 1:3 and 1:5 is quite sensible, traditional, and powerful
(in terms of theory and explanatory adequacy). I look forward to an
article using, say, 1:3 and 3:5, but I think 1:5 is a better choice.

Let's look at 17. Assuming a starting point of D, the 5-unit m3 is F/.
What's wrong with that? Of course this is also F#\, reflecting the
important fact that this tuning contains a neutral third. In other words,
the M3 and m3 meet at 5 units. If you want a 282-cent m3, because it is a
few cents closer to the just 5:6, that's fine; it may be called F.

There is no more inherent reason to expect the m3 and M3 to "behave
properly" than there is for cycles of P5 or for the M3 and P5 together.
Some tunings, like 25 and marginally like 17, may have two minor thirds.
Tunings with large numbers of units in the octave may have several of
everything. I don't find this a fault of notation in general or of my
proposal.

In 20 there is the curious result that all the extant commas are an even
number of units, but it doesn't matter.

> Now I noticed there is a part of your paper where you give an example of
> defining a comma to be other than its "best" size, and show the problems
> that entails for notation. Though you refer to the alternate comma
> definition as the "disposition" of the tuning, as if it were a property of
> the tuning itself, one might infer from that discussion that the "best"
> comma size is the one to use. However, for tunings like 20tET which do not
> possess 5-limit consistency, this is not a clear-cut issue.
>
Tunings have many properties, all of which cannot be most clearly shown
at once. The discussion does not imply that the "best" size always works.
Quite the contrary. That is one point of the article. In some cases I
am unsure what to do about it. I intend a follow-up article at some time.

> The other commas you use derive from equivalences in 12tET, and therefore
> seem undesirable for a progressive microtonal notation.

In one sense this is true, viz. that the Pythagorean komma is 12v-7a,
where v is a P5 and a is an octave. That governs a lot, as that komma
figures in the others. But in another sense it's irrelevant, because I
have attempted to maintain some conventions, including heptatonic naming,
which are based on pre-12 considerations, as Paul E. shows, all in a 5-limit.
But I don't "measure" all tunings "in reference to 12-tET."

> In your notation, the syntonic comma is a natural expansion of the standard
> notation, so that the entire 5-limit just system can be expressed in terms
> of the Pythagorean scheme above. This works fine for tunings that are
> consistent within the 5-limit. But the other commas are measured as the
> difference between three major thirds and an octave, and so notating them as
> an alteration, rather than an essentially different note, implies an
> underlying dodecaphony, much as standard notation implies an underlying
> heptatonicism.

I don't think so. It implies setting up the 3-limit to take care of itself
(sharps/flats) and then seeing what happens with the 5-limit intervals in
terms of the octave, as Paul E. states. There's no 12 in there. By the
way, I am quite interested in nonstandard naming too, as a previous XH
article on 13-tET shows. Would Paul E. elaborate on notating something with
different notes? It's hard to see how he gets from commas to notes in the
quotation above.

What is "consistent within the 5-limit"? He may have something because,
on occasion, use of kommas others than syntonic seems a bit arbitrary. In
some cases it's fine, as in 31.

> Our present notation system was established long before 12tET became
> standard. If 19tET or 31tET had become standard instead, your basic set of
> non-syntonic commas would be quite different. If the practice of notating
> and differentiating syntonic commas in music had become standard, your set
> of other commas might have derived from equivalences in 22 or 34tET. You
> would have to argue that your set of commas comes solely from a
> consideration of the simplest small intervals in just intonation to justify
> this apparant reliance on 12tET equivalences.

In a sense this is true, with reference to the 12v-7a I mentioned
earlier. But that is possibly a simpler comma than 19v-11a, 31v-18a,
22v-13a, etc. Maybe we should go for 7v-4a?

The interesting question then becomes how do you build a general system,
let us say still on a 5 limit, without this one bit of 12 in it--a system
which is no more tied to anything than this one is?

> A final quibble: A positive tuning is customarily defined as one where g# is
> higher than a-flat, i.e., where the perfect fifth is larger than the 12tET
> fifth, not the just fifth as you state. Thus the sign of the pythagorean
> comma is the determining factor here, and so the customary definition would
> fit better into your own scheme.

This isn't a quibble, it's an outright error. It has been pointed out
before, and quite honestly I don't know how I let it get in there. Paul E.
is correct.

> You illustration of 171tET notation is interesting. A simpler tuning that
> has consistent and unique representations through the 16th harmonic is
> 111tET. For example, the sequence 16:15,15:14,14:13,13:12,12:11,11:10 is
> represented by 10,11,12,13,14,15 steps in 111-tET

This bears consideration, I believe.

I am grateful to Paul Erlich for his thorough consideration of my
article. Put in his terms, I can see the bit of 12 behind some of my
proposals. They come from other considerations, mostly, which are not
12-related. In any case, the challenge is always there for someone to do
something better, or at least as thorough.

========================== =================================
Dr. Paul Rapoport e-mail: rapoport@mcmaster.ca
SADM (Music) tel: (+1) 905 529 7070, ext. 2 4217
McMaster University fax: (+1) 905 527 6793




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