>As Paul kindly said, at least with fifths it's a manageable sort of
>mess. :-) And I would add: with many familiar landmarks, particularly
>in the harmony.

This sort of attitude vastly:

() Underestimates the effect of notation on music. Use ordinary
notation, think up ordinary music.

() Overestimates the difficulty of 'learning new nominals'. You've
got new pitches, new rules, new fingerings, new sounds, new accidentals.
The nominals matter so much?

>> >No. A little reflection allows me to explain that, as it stands now,
>> >the semantic foundations of Sagittal notation have absolutely
>> >nothing to do with any temperament.
>>
>> I should have said, "good PBs" there. [I think of PBs as
>> temperaments, which always gets me into trouble.]
>
>So what's a _good_ PB for notational purposes?

The same kind that are good for composition purposes!

>That sounds even less
>likely to be agreed upon than a good linear temperament. How about
>we forget about this given our agreement below?

I thought it was well-agreed-upon: the simplicity of the commas vs.
their size. There are different ways to calculate this, and the details
of how to do so with planar and higher temperaments and raw PBs are not
settled, but using any of the proposed methods is fine -- pick your fav.

>> With linear temperaments, you only need 1 accidental pair at a time,
>> as I've pointed out.
>
>But Carl, that's like saying you only need 6 pairs of accidentals to
>notate 19-limit JI. One for each prime above 3. It becomes essentially
>unreadable once you go past 2 accidentals per note.

How is saying you only need 1 like saying you only need 6?

>And even ignoring these "enharmonics", you need other accidentals when
>you have multiple parallel chains, i.e. when the period is not the
>whole octave.

Isn't this refuted by Paul's single-accidental decatonic notation?

>> If average use ("gimme 9 notes of such-and-such temperament in the
>> 13-limit") turns out to require more commas than can fit on a list,
>
>I don't understand how average-use could require "more commas than can
>fit on a list". What could this mean except "an infinite number of
>commas"?

I thought you said something about the list getting unwieldy. If
you've come up with 600 symbols, I think that should be plenty!

>> you could try assigning (an) accidental(s) for each *temperament*,
>> with the understanding that it/they would take on TM-reduced value(s)
>> for the limit and scale cardinality being used.
>
>Eek! So then we would have to learn not only new nominals for every
>temperament, but new accidentals too?

Instruments don't read accidentals; people do. I'm not sure how
learning 600 accidentals is any easier than learning tuning-specific
interpretations of existing accidentals. In both cases, once the
tonal system is learned, one should be able to hear the correct notes.
And this proposal has the added benefit of not requiring any new fonts
or eye training -- just use conventional sharps and flats.

It's just a proposal. Drawbacks include:

() Only works for linear temperaments.

() It's kinda neat to not have to specify the temperament in advance.
One could mix "temperaments" in the same bar just by using the
appropriate accidentals from a master-list. Can't do this with the
present proposal.

>There's definitely no need for this.

How do you know? Who can say what composers won't need?

>> >So the first part of my belief is that it is far better to have a
>> >notation system whose semantics are based on precise ratios and then
>> >use that to also notate temperaments, rather than trying to find the
>> >ultimate temperament and then using a notation based on that to notate
>> >both ratios and other temperaments.
>>
>> Wow; this is exactly what I've been saying all along!!
>
>Really? Then how have I managed to waste so much of my time answering
>this thread?

Glad to see you have such a high opinion of peer review.

>> >Then if that's accepted, the second part is that it is best if the
>> >simplest or most popular ratios have the simplest notations.
>>
>> Right. And it's this aspect that makes the search more-or-less
>> equivalent to the search for good PBs.
>
>Nope. You've lost me there.

The simplest commas would be the most popular for a reason!

>> >I understand that you agree with this, and so it should be obvious
>> >that the simplest accidental is no accidental at all and so the
>> >simplest ratios should be represented by nominals alone. When we
>> >agree that powers of 2 will not be represented at all, or will be
>> >represented by an octave number, or by a distance of N staff positions
>> >or a clef, then surely you agree that the next simplest thing is to
>> >represent powers of three by the nominals.
>>
>> Well, that's a weighted-complexity approach. But even with most
>> weighted-complexity lists I've seen, non-rational-generator
>> temperaments appear.
>
>Huh? I thought you just agreed that we would first decide how to
>_precisely_ notate ratios?

Yup. In fact, you can think of a PB/temperament *as* a notation in
my scheme.

>Therefore we don't care about weighted
>complexity, or any complexity (except at the 3-prime-limit), because
>we know we are going to represent ratios of the other primes as being
>_OFF_ the chain, by using accidentals.

Don't follow you here. But try to track me again. By saying you
want to always keep the lowest primes the simplest ones in the map
(by assuming 2-equiv. on the staff and by always using 3:2s for your
nominals), you are effectively weighting your complexity measure.
If you completely disallow temperaments like miracle (which do not
have a 3:2 generator) from showing up in your notation search (think
temperament search), it's a *very* strongly weighted function --
you're insisting that both generators be primes.

>Whether we use rational or irrational generators we can only represent
>powers of _ONE_ ratio _EXACTLY_, _ON_ the chain, (modulo our interval
>of equivalence).

"Ratio" obviously. Did you mean "prime"? Then your statement is false.

>> But certainly the project didn't start out this way, and even in
>> the last few days I saw a blurb for George and/or you looking very
>> confused about non-heptatonic systems.
>
>I think we're only confused about how a notation whose nominals are
>related by an irrational generator could be used notate ratios
>precisely.

Just observe Paul's decatonic notation. It's the perfect embodiment
of everything I've been saying.

-Carl