[tuning-math] Re: A common notation for JI and ETs
- From: "gdsecor" <gdsecor@xxxxxxxxx>
- To: tuning-math@xxxxxxxxxxxxxxx
- Date: Wed, 14 Aug 2002 18:07:04 -0000
--- In tuning-math@xxxx, David C Keenan <d.keenan@xxxx> wrote:
> At 11:52 13/08/02 -0700, George Secor wrote:
> >From: George Secor, 8/13/2002 (tuning-math #4577)
> >Subject: A common notation for JI and ETs
> >
> > ... And now that I've taken a fresh
> >look at the notation, I came up with some ideas on how to improve
a few
> >things.
> >
> >First of all, here is how I was able to notate all of the 15-odd-
limit
> >consonances taking C as 1/1. (Don't bother to look through all of
this
> >now; I'll be referring to many of these below, so this listing is
just
> >given for reference.)
> ...
>
> That's marvellous, except of course it looks like gobbledygook when
up to 5 ASCII symbols are being used to represent a single sagittal
symbol. How big is the biggest schisma involved?
As best I can remember there was nothing significantly more than 1
cent. (I'll be reporting most of them as I go along.) I would like
to have the 217-tone symbols used for JI only as a last resort, which
enables us to keep the 15 limit notation system-independent.
> >> I don't think we have defined a rational complement for /|~
because
> >it isn't needed for rational tunings.
> >
> >On the contrary, I found that /|~ is in fact quite useful for
rational
> >tunings (see above table of ratios), but its lack of a rational
> >complement is a problem. To remedy this, I propose ~||( as its
> >rational complement.
>
> Fair enough, and yes, that would seem the obvious complement.
>
> >With C as 1/1, the following ratios would then
> >use these two symbols (which also appear in the table of ratios
above):
> >
> >11/10 = D\!~
> >20/11 = Bb/|~ or B~!!(
> >15/11 = F/|~
> >13/7 = B\!~
> >14/13 = Db/|~ or D~!!(
> >
> >In effect, /|~ functions not only as the 5+23 comma (~38.051c), but
> >also as the 11'-5 comma (~38.906c) and the 13'-7 comma (~38.073c)
>
> OK, so a 0.86 c schisma. I can certainly live with that for such
obscure ratios.
>
> >This would replace (|( <--> ~||( as rational complements. I found
that
> >(|( is not needed for any rational intervals in the 15-odd limit,
so
> >this has no adverse consequences. (However, it leaves the 23'
comma
> >without a rational complement; I will address that problem
below.) The
> >new pair of complements that I am proposing also has a lower offset
> >(0.49 cents) than the old (-1.03 cents), so, apart from the 23'
comma,
> >I can't think of a single reason not to do this.
>
> Me neither. Apart from the 23' comma.
>
> We could resurrect ~)||, with two left flags, as the complement of
the 23' comma. It isn't like a lot of people really care about ratios
of 23 anyway. We already made a good looking bitmap for ~)| with the
wavy and the concave making a loop.
I'll be addressing this later.
> >The reverse pair of complements, ~|( <--> /||~, would be used for
the
> >following ratios of 17:
> >
> > 17/16 = Db~|( or D\!!~
> > 17/12 = Gb~|( or G\!!~
> > 17/9 = Cb~|( or C\!!~
> > 32/17 = B~!(
> > 24/17 = F#~!( or F/||~
> > 18/17 = C#~!( or C/||~
> >
> >All of this is going to affect how we will want to notate not only
152,
> >but also other ETs, including 217. (More about this later.)
>
> If rational complements don't have to be consistent with 217-ET any
more, how about making rational complements consistent with 665-ET,
as proposed earlier?
And I'll answer this one at the same time as the previous, because I
believe they're related.
> >... The principle that I am advancing here is that there is
another goal or
> >rule that should take precedence over that of an easy-to-memorize
> >symbol sequence -- symbols which are used to represent JI
consonances
> >should be used in preference to those that can be expressed only as
> >sums of comma-flags. These are the symbols that will be used for
JI
> >most frequently, and they will therefore (through repeated use)
become
> >*the most familiar* ones.
>
> But many people using ETs couldn't care less about JI, so why
should rational approximations take precedence over mnemonics,
particularly if they only involve ratios as uncommon as 5:11 and 7:13?
I think you meant 15:11, because I was going to remark: Are there
really so few who would venture beyond the 7-limit? (But you are
still going to encounter 15:11 in the 11 limit.) My experience is
that two things take place the longer you are into microtonality:
1) If you use temperaments, you tend to prefer systems with less
error in the intervals than you did at first; and
2) You are able to accept (or find use for) a higher harmonic limit.
When I performed some of Ben Johnston's music in the mid '70s he was
composing in 5-limit JI, but he didn't stop there. Given enough
time, I think that you're going to find 15-limit ratios becoming more
and more common.
Anyway, my objective is to *minimize* the total number of symbols
that are likely to be encountered by performers, who are likely to
be involved with *both* JI and ETs. Fewer symbols would, in turn,
decrease the possibility of misreading or confusing them and would
also make the process of memorization easier. Remember, this is
supposed to be a *common* notation for JI and ETs, and having symbols
in the ETs that are even more uncommon (i.e., both different and
infrequent) than 15-limit JI ones (which is what occurs much of the
time when you try to minimize the wavy and concave flags) tends to do
the opposite.
Let's pass judgment on this after we've looked at how this works out
with a number of ETs.
> >And these are the symbols that should have
> >first priority in the assignment of rational complements.
>
> Yes. I can accept that.
>
> > This is why
> >I want to eliminate (|( in the rational complement scheme -- it is
the
> >(13'-(11-5))+(17'-17) comma or, if you prefer, the (11'-7)+(17'-17)
> >comma, neither of which is simple enough to indicate that it would
ever
> >be used to notate a rational interval; and none of the 15-limit
> >consonances (relative to C=1/1) require it.
>
> I'll wait and see where this leads. By the way, I assume we agree
that many of those 15-limit "consonances" are not consonant at all,
and are not even Just, being indistinguishable from the intervals in
their vicinity, except if they are a subset of a very large otonality
or with the most contrived timbre.
Try mistuning an 11:13:15 triad -- you will hear the combinational
tones beat against one another. As I understand it, this is the
essence of JI.
> >This will be continued, following a short digression about 76-ET.
> > ...
> If we are proposing a _single_ standard way of notating every ET
then 76 should be as a subset of 152-ET. However I think there are
several such ETs where some composers may have very good reasons for
wanting to notate them based on their native best fifth, (for example
because the 76-ET native fifth is the 19-ET fifth), and we should
attempt to standardise those too. So I say give both, but favour the
152-ET subset.
Okay, that makes sense!
--George
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