[tuning-math] Re: A common notation for JI and ETs
- From: "gdsecor" <gdsecor@xxxxxxxxx>
- To: tuning-math@xxxxxxxxxxxxxxx
- Date: Thu, 15 Aug 2002 19:28:14 -0000
--- In tuning-math@xxxx, David C Keenan <d.keenan@xxxx> wrote:
> At 11:27 14/08/02 -0700, George Secor wrote:
> >Until recently I had a prejudice against //|, because it has two
flags
> >on the same side. But now that I see that other symbols of this
sort
> >haven't popped up all over the place, and since its rational
complement
> >~|| is simple and useful, I would like to include it in the
standard
> >217 notation instead of ~|\ (which is only the 11-5+17 comma, and
which
> >is not needed for any 15-limit consonances).
>
> That's fine by me. I totally approve of making more use of //|, but
it should only be used in an ET if it is valid as the double 5-comma.
Yes, a mandatory test for the use of this symbol in an ET is that the
ET be 1,5,25 consistent.
> >For reference, here is the 217 standard notation as it presently
> >stands:
> >
> >217: |( ~| |~ /| |) |\ ~|) ~|\ /|) /|\ (|) (|\ ~||
||~ /|| ||) ||\ ~||) ~||\ /||) /||\ (present)
> >
> >Making this change would give us:
> >
> >217: |( ~| |~ /| |) |\ ~|) //| /|) /|\ (|) (|\ ~||
||~ /|| ||) ||\ ~||) //|| /||) /||\ (all RCs)
> >
> >So we would now have true rational complements throughout.
> >
> >However, there is a second change that I wish to propose. It
> >incorporates the change of rational complements from (|( <--> ~||(
to
> >/|~ <--> ~||( that I also proposed above. For 7deg we now have
~|),
> >which is used for the following ratios, but for nothing in the
> >15-limit:
> >
> >17/10 = Bbb~|) or Bx~
> > 17/15 = Ebb~|) or Ex~
> >
> >(For this ascii notation I have used x instead of X to specify a
> >*downward* alteration of pitch, as we have already done with !
instead
> >of |. I hope the presence the wavy flag in combination with it is
> >enough to indicate it is not being used here to indicate a double
> >sharp. Otherwise, would a capital Y be a suitable alternative?)
>
> Little x for downward is fine with me.
>
> >The proposed replacement standard symbol /|~ for 7deg217 is used
for
> >11/10, 14/13, and 15/11 (plus their inversions).
> >
> >In order to maintain rational complements and a matching symbol
> >sequence throughout, the symbols for 3, 14, and 18deg217 would also
> >need to be changed, which would give this for the standard 217 set:
> >
> >217: |( ~| ~|( /| |) |\ /|~ //| /|) /|\ (|) (|\ ~||
~||( /|| ||) ||\ /||~ //|| /||) /||\ (new RCs)
> >
> >The 3deg symbol changes from the 23 comma (or 19'-19 comma, if you
> >prefer) to the 17' comma. This is a more complicated symbol, but
it
> >symbolizes a lower prime number, making it more likely to be used.
> >(Besides, it has mnemonic appeal.)
>
> Yes I suppose I can give up monotonic flags-per-symbol, but if you
don't want to know about JI or don't care about 11/10, 14/13, or
15/11, then that /|~ now seems to come out of nowhere. Why suddenly
introduce the right wavy flag. At least ~|) introduces no new flags.
Three reasons:
1) As I said above, /|~ is used for 3 15-limit ratios (not including
inversions), while ~|) is used for only one ratio of 17. Hence /|~
will have a wider use.
2) Those who don't care about 11/10 _et al_ will probably be using
tempered versions of these ratios in one way or another if /|~ occurs
in the particular ET they are using. Use of the same symbol in
*both* JI and the ET exploits the *commonality* of the symbols for
both applications.
3) As I said below, I am now placing a higher priority on minimizing
the number of the most commonly used *symbols* than on minimizing the
number of *flags* used for an ET. This "most commonly used" set of
symbols was summarized in the 8 sets of rational complements that I
listed at the end of my last message.
> >My goal is to minimize the differences between the 217-ET notation
and
> >the rational notation (while maintaining a matched symbol
sequence),
> >with the lowest primes (i.e., the 17 limit) being favored.
>
> That's fine so long as it is the 217-ET notation that gets
compromised, not the rational.
>
> >This would
> >make the transition from purely rational symbols to 217-ET standard
> >symbols as painless as possible in instances where the composer
has run
> >out of rational symbols and has no other choice but to use 217
symbols
> >to indicate rational intervals.
>
> I don't understand why there would be no choice but 217-ET. Is 217-
ET really the best ET that we can fully notate? What about 282-ET?
It's 29-limit consistent. I've never really understood the deference
to 217-ET.
I never considered 282 before, but I do see some problems with it:
1) 11 is almost 1.9 cents in error, and 13 is over 2 cents; these
errors approach the maximum possible error for the system. (This is
the same sort of problem that we have with 13 in 72-ET.)
2) The |) flag is not the same number of degrees for the 7 and 13-5
commas (which is by itself reason enough to reject 282), nor is (|
the same number of degrees for the 11'-7 and 13'-(11-5) commas.
3) The following rational complements for the 15-limit symbols are
not consistent in 282:
)|~ <--> (|| 19' comma
|( <--> /||) as 7-5 comma or 11-13 comma (but 17'-17 is okay)
~| <--> //|| 17 comma
|) <--> ||) 7 comma
//| <--> ~|| 25 comma
(| <--> )||~ 11'-7 comma
And besides this, there are others that are inconsistent, such as:
|~ <--> ~||) as both the 19'-19 and 23 comma
What makes 217 so useful is that *everything* is consistent to the 19
limit, and, except for 23, to the 29 limit. And I think that the
problems with 23 can be managed, considering how rarely it is likely
to be used. You have to have a way to accommodate the electronic JI
composer who might want to modulate all over the place, and a
consistent ET mapping for JI intervals is the only way to do it with
a finite number of symbols; this is where 217 really delivers the
goods!
> ...
> >So I think it would be best to retain the straight flags in the
> >standard 217 set,
>
> Agreed.
>
> > but to have in mind the (| and )||~ symbols as
> >supplementary rational complements. A composer would have the
option
> >to use (| and )||~ to clarify the harmonic function of the tones
which
> >they represent for either 217-ET or JI mapped to 217. The same
could
> >be said for the rational symbols for ratios of 19 and 23, should
one
> >want to use a higher harmonic limit. (These would be less-used,
> >less-familiar symbols that would be rarely be needed below the 19
> >limit.)
> >
> >With these changes in the standard 217 notation, it would be
necessary
> >to memorize only 8 rational complement pairs (half of which use
only
> >straight and convex flags, and half of which are singles, not
pairs) to
> >notate all of the 15-limit consonances and a majority of the
ratios of
> >17 in JI:
> >
> >5 and 11-5 commas: /| <--> ||\ and |\ <--> /||
> >7 comma: |) <--> ||)
> >11 diesis: /|\ <--> (|)
> >13 diesis: /|) <--> (|\
> >7-5 comma or 11-13 comma: |( <--> /||)
> >17 comma and 25 comma: ~| <--> //|| and //| <--> ~||
> >17' comma and 11'-5 or 13'-7 comma: ~|( <--> /||~ and /|~ <--> ~||
(
> >19' comma and 11'-7 comma: )|~ <--> (|| and (| <--> )||~
> >
> >(The last pair of RCs are the supplementary symbols that are not
part
> >of the standard 217-ET set.)
> >
> >With these symbols you have more than enough symbols to notate a
> >15-limit tonality diamond (with 49 distinct tones in the octave).
>
> Good work. I'd like to see that listed in pitch order.
At first I thought you meant listing the symbols like this:
Symbol set used for 15-limit JI
-------------------------------
)|~ <--> (|| 19' comma (not in standard 217 set)
|( <--> /||) 7-5 comma or 11-13 comma
~| <--> //|| 17 comma
~|( <--> /||~ 17' comma
/| <--> ||\ 5 comma
|) <--> ||) 7 comma
|\ <--> /|| 11-5 comma
(| <--> )||~ 11'-7 comma (not in standard 217 set)
//| <--> ~|| 25 comma
/|~ <--> ~||( 11'-5 or 13'-7 comma
/|\ <--> (|) 11 diesis
/|) <--> (|\ 13 diesis
But now I think you meant listing the ratios like this:
Sagittal Notation for 15-limit JI
---------------------------------
1/1 = C
16/15 = Db/| or D!!/
15/14 = C#|( or C|||(
14/13 = Db/|~ or D~!!(
13/12 = D(!/
12/11 = D\!/
11/10 = D\!~
10/9 = D\!
9/8 = D
8/7 = D|)
15/13 = D//|
7/6 = Eb!) or E!!!)
13/11 = Eb!( or E!!!(
6/5 = Eb/| or E!!/
11/9 = E(!)
16/13 = E\!)
5/4 = E\!
14/11 = Fb(| or F)||~
9/7 = E|)
13/10 = F\\!
4/3 = F
15/11 = F/|~
11/8 = F/|\
18/13 = F(|\
7/5 = Gb!( or G!!!(
10/7 = F#|( or F|||(
13/9 = G(!/
16/11 = G\!/
22/15 = G/!~
3/2 = G
20/13 = G//|
14/9 = Ab!) or A!!!)
11/7 = G#(! or G)||~
8/5 = Ab/| or A!!/
13/8 = A(!/
18/11 = A\!/
5/3 = A\!
22/13 = A|(
12/7 = A|)
26/15 = Bb\\! or B\\!!!
7/4 = Bb!) or B!!!)
16/9 = Bb or B\!!/
9/5 = Bb/| or B!!/
20/11 = Bb/|~ or B~!!(
11/6 = B(!)
24/13 = B\|)
13/7 = B\!~
28/15 = Cb!( or C!!!(
15/8 = B\!
2/1 = C
> >Notice that I identified |( as something other than the 17'-17
comma.
> >This is because it is used for the following rational intervals:
> >
> >7/5 = Gb!( or G!!!(
> >10/7 = F#|( or F|||(
> >13/11 = Eb!( or E!!!(
> >22/13 = A|(
> >15/14 = C#|( or C|||(
> >28/15 = Cb!( or C!!!(
> >
> >Thus |( can assume the role of either the 17'-17 comma (288:289,
> >~6.001c), the 7-5 comma (5103:5120, ~5.758c), or the 11-13 comma
> >(351:352, ~4.925c). However, there are a limited number of ETs in
> >which it can function as all three commas (159, 171, 183, 217, 311,
> >400, 494, and 653) or at least as both the 7-5 and 11-13 commas
(130
> >and 142).
>
> Hmm. It is certainly arguable that we should favour the
interpretation of |( as the 7-5 comma when notating ETs. What's the
smallest ET that would be affected by this?
It's hard to say what is the smallest ET in which they differ
consistently. We're looking at three different commas: 7-5, 11-13,
and 17'-17, and we're dealing with all of the primes in the 17
limit. All three commas are the same number of degrees (without
vanishing) in 19, 43, 159, 171, 183, 217, 311, 400, 494, and 653.
All three are a different number of degrees in 612 and 1600. 7-5 and
11-13 are like degrees but differ from 17'-17 in 26, 60, 72, 84, 96,
130, 142, 176, 224, 270, 282, and 364. The 7-5 and 11-13 agreement
is important for rational complementation, because it is the number
of degrees in the 11-13 comma that determines whether |( <--> /||) is
consistent, whereas the 17'-17 agreement is important only if the
notation for a given ET also uses both the 17 and 17' symbols. The
ETs above 100 that I looked at in which these don't agree are 108,
118, 120, 125, 132, 144, 147, 149, 152, 193, 207, 388, 525, 612, 742,
and 1600; many of these won't even need the |( symbol, and 193 is
probably the most important one in which |( would be used that is not
consistent with the 7-5 comma (although the 19 comma could be used
instead with /||) as its alternate complement; another alternate
complement is already required for the 17 comma of 2deg, so this
doesn't harm a notation that might otherwise have all rational
complements and a matching sequence).
> Is )| still to be interpreted as the 19 comma and what is to be its
complement?
Yes, and its complement is still (||~. I don't see any lower-prime
interpretations of it without going into rational complements, where
we have only one: 11/7 = G)||~. This is greater than G(|)
(2187/1408) by 15309:15488, ~20.125c (vs. the 19' comma, 19456:19683,
~20.082c). But this is for )|~, so we must subtract |~ from this,
but what comma would |~ be? Since your next question has a positive
answer (and since I did that one first I can peek at the answer),
I'll use the 11-limit comma 99:100, which gives 42525:42592
(3^5*5^2*7:2^5*11^3, ~2.725c) as the 11-limit interpretation of )|.
This is meaningful only if you are using rational complements, i.e.,
single-symbol notation.
> Is there a lower prime interpretation of |~ now too?
Hmm, good question! Yes, using /|~ as the 11'-5 comma for 11/10
would make that symbol 44:45, so |~ would be 99:100, ~17.399 cents.
And using /|~ as the 13'-7 comma for 13/7 would make /|~ 1664:1701,
so |~ would be 104:105, ~16.567 cents. ^
> It seems to me that what we are discussing here is unlikely to
impact on many ETs below 100. Is that the case?
Yes, I think that this will affect mostly the weird and difficult
ones. We have been able to do the simpler ones using only straight
and convex-right flags, which have remained unchanged.
--George
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