This email from George might be useful:
Subject: Re: New comma popularities
--- Dave Keenan <firstname.lastname@example.org
> At 07:45 AM 8/06/2005, [George] wrote:
>> Sorting it all out takes time, and we can't rush through this as if
>> we're on a deadline.
> You're right. Sorry.
>> As a result, I've come up with a procedure that, so far, arrives at the
>> entire athenian set (both the symbols and their primary roles, without
>> skipping over any comma-ratios or any other exceptions), with the only
>> "givens" being the comma-popularity list, the Sagittal flags and
>> accents, and athenian-level JI defined as a division of the apotome
>> into 21 (more or less equal) parts.
> Sounds brilliant.
It isn't really difficult at all, if you're willing to accept (and
enforce) a one-to-one correspondence between symbols and their primary
roles, such that any symbol will always have the same primary role
(regardless of JI resolution) and a comma-ratio can be the primary role
of one (and only one) symbol. That, taken together with a rule that
new symbol cores should *not* be introduced if a left-accent applied to
an existing symbol core (not previously disqualified from being
athenian-level) will notate a ratio *exactly*, pretty much does the
trick. This has the consequence of maximizing
drop-accents-for-lower-resolution (DAFLR, my latest, and hopefully now
best-and-only acronym for this property) from olympian to herculean to
This may also produce some unintended or unexpected consequences as I
get farther down the popularity list, but that remains to be seen.
I've found that the results so far tend to favor the creation of
monotonic core sequences, but I haven't gotten far enough along yet to
formalize any rules for that, other than prohibiting the creation of
left-accented symbols that would result in the unaccented and accented
symbol pair ending up on opposite sides of an unaccented athenian-cored
You had earlier said that you would like to avoid having to go through
olympian mina by mina, deliberating over what should be each symbol and
primary role. I'm finding that most of them are pretty obvious and
that there should only be a handful or so left to debate -- not
that a handful won't cause us a lot of trouble. ;-(
>> It also simultaneously derives
>> herculean, promethean, and olympian symbols and their primary roles as
>> far as I've taken it, and I'm hoping that a continuation of this
>> procedure will bring it to completion. I'm having to determine some of
>> the details of the procedure as I go along, but so far (through 245S)
>> it's looking *extremely* good.
> That's exciting.
>> I've reached a point where I have a question that's stopped me in my
>> tracks: How did the next comma, 23S (16384:16767), get counted as more
>> popular than 23C (729:736)? The size of the numbers in the ratio alone
>> looks suspicious.
>> With C=1/1, 23/16 may be spelled either F#|~ or
>> Gb~|\, which means we're altering a nominal 6 places in a chain of 5ths
> >from C in each case. Looking at it from a composer's perspective, I
>> would probably want to spell it as F-something if I were treating it as
>> a dissonance, which I would most likely resolve by moving up to G.
>> As I see it, everything points to having 23C appearing above 23S, so
>> please explain how you got this.
> The comma complexity is misleading. As you say, the absolute exponent of 3
> in both cases is 6. You get the different comma complexities because in one
> case the 23 is on the same side of the ratio as the 3^6 which means it
> takes a larger power of two to counteract it.
Yes, but that's 23S, not 23C.
> You will remember that "slope" is the rate of change of comma size with
> notational fifth size when the comma size is expressed as a fraction of the
> apotome. It's a measure of how useful a comma is likely to be in notating
> temperaments as well as rational tunings.
> 23S has the 3^6 on the larger side of the ratio and the apotome has 3^7 on
> its larger side, which means that 23S will remain almost the same fraction
> of the apotome as the notational fifth varies in size, while 23C goes the
> other way more rapidly.
> As you can see in the comma-popularities spreadsheet, 23S has an absolute
> slope of 3.5 while 23C has 7.0. So 23C is not as useful notationally
> because it will tend to change its position too wildly relative to other
That's nice to take into consideration when you're notating
temperaments, but what relevance does it have to notating JI?
> If you can think of a simple formula that can take into account knowledge
> as seemingly specific as "... spell it as F-something if I were treating it
> as a dissonance, which I would most likely resolve by moving up to G", then
> please let me know.
> Maybe it's just that relative to C as 1/1 we like to have our "compound
> nominals" in the range of Ab to G# (-4 to +8 fifths), so F# is preferable
> to Gb. I could adjust the formula to give that some weight.
Did you intend to have both Ab and G# in that range, or did you mean Eb
to G# or possibly Ab to C#?
Db should also be preferable to C# (for reasons relating to 17/16), so
I think that range should be Db to F#. Yes, you could try something
> How is this actually impacting on your results (so I can figure out how
> much I need to adjust things to get the results you want)? e.g. Tell me
> what other comma(s) you need 23C to be more or less "popular" than.
I'll have to work down the list through 23C before I know. It's very
possible that there may not be any impact at all.
>> I said above that it doesn't use a strict EDA scheme.
> Yes. But even so, it is useful to have an EDA as a reference when
> interpreting this stuff.
>> My objective is
>> to a set of unaccented symbols that will convert more smoothly between
>> herculean and promethean (!!!) than those in 42 or 47-EDA, and I've
>> concluded that it doesn't necessarily have to be an EDA to do that.
> I agree it doesn't have to be an EDA. But it shouldn't be too uneven.
I've found that there tends to be less unevenness in the region from |(
to /| than outside that.
>> (or fielding questions about what's linear or
>> MOS, for that matter).
> A silly discussion.
It seems to be pretty important to Paul. He's put a lot of time and
effort into the classification of temperaments, and he doesn't want to
be shot down by somebody out there in academia accusing him of sloppy
terminology, and I imagine he's getting rather frustrated by a lack of
progress in building any sort of consensus. You don't have to look any
farther than some the "silly" details we've be discussing the past year
or two to appreciate that.
Anyway, I'll keep you up to date on my progress as the popularity
decreases, and perhaps toss you some more specifics about the rules I'm