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Message: 5126 - Contents - Hide Contents

Date: Wed, 31 Jul 2002 02:07:15

Subject: Four 10-note, 7-limit JI scales

From: Gene W Smith

If we take (10/9)^2 (15/14)^2 (16/15)^2 (21/20)^3 = 2 as scale steps, and
simplify the scale-finding problem by assuming 4/3 and 3/2 both belong to
the scale, we obtain four scales, the third and fourth of which are the
inverted forms of the first and second. A version of the major/minor
transformation, exchanging 10/9 with 16/15, which is equivalent to saying
2-->2, 3-->3, 5-->24/5, 7-->168/25, exchanges the first and second, as
well as the third and fourth. The first "decaa", and fourth, "decad", are
major versions, having two major tetrads and a minor tetrad, while
"decab" and "decac" have two minor and one major tetrad. In any system
where 50/49~1 the exchange transform sends tetrads to tetrads and can be
considered major/minor. In 22-et in particular, each scale becomes the
symmetrical decatonic. All of the scales have 23 intervals, 17 triads and
3 tetrads.

! decad.scl
! [15/14, 10/9, 21/20, 16/15, 15/14, 21/20, 10/9, 15/14, 16/15, 21/20]
inversion of decab
10
!
15/14
25/21
5/4
4/3
10/7
3/2
5/3
25/14
40/21
2/1

! decab.scl
! [21/20, 16/15, 15/14, 10/9, 21/20, 15/14, 16/15, 21/20, 10/9, 15/14]
(10/9) <==> (16/15) transform of decaa
10
!
21/20
28/25
6/5
4/3
7/5
3/2
8/5
42/25
28/15
2/1

! decac.scl
! [15/14, 16/15, 21/20, 10/9, 15/14, 21/20, 16/15, 15/14, 10/9, 21/20]
inversion of decaa
10
!
15/14
8/7
6/5
4/3
10/7
3/2
8/5
12/7
40/21
2/1

! decad.scl
! [15/14, 10/9, 21/20, 16/15, 15/14, 21/20, 10/9, 15/14, 16/15, 21/20]
inversion of decab
10
!
15/14
25/21
5/4
4/3
10/7
3/2
5/3
25/14
40/21
2/1


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Message: 5127 - Contents - Hide Contents

Date: Wed, 31 Jul 2002 23:34:24

Subject: Tempered versions of Carl's 12-note JI scales

From: Gene W Smith

I took these five scales and looked for what new 11-limit intervals would
appear if we allowed commas of less than 8 cents. This puts us in a range
well covered by the 72-et, but none of the resulting scales needed the
full power of this, or even of miracle; they all were covered by one or
another planar temperament. Except for the case of "lester", which is in
the {225/224, 441/440} temperament which doesn't improve from the 72-et
values, this meant some improvement in the tuning was possible. Whether
it is worthwhile is another question.

# lumma in {385/384, 441/440} temperament, 873-et version
l873 := [873, 1383, 2026, 2449, 3019];

lum:=[0, 37, 170, 230, 280, 400, 450, 510, 643, 680, 740, 813];
lumd := [37, 133, 60, 50, 120, 50, 60, 133, 37, 60, 73, 60];

42 ingervals, 58 triads--the least harmony, but the best tuning

# {225/224, 385/384} 858-et version
l858 := [858, 1359, 1990, 2408, 2967];

# prism

prs := [0, 83, 144, 191, 274, 357, 418, 501, 584, 631, 692, 775];
prsd := [83, 61, 47, 83, 83, 61, 83, 83, 47, 61, 83, 83];

49 intervals, 86 triads--the champ. It's also fairly regular.

# stelhex

ste := [0, 61, 191, 227, 274, 335, 418, 501, 584, 645, 692, 728];
sted := [61, 130, 36, 47, 61, 83, 83, 83, 61, 47, 36, 130];

46 intervals, 72 triads

# class

cla := [0, 61, 108, 227, 274, 335, 418, 501, 548, 645, 692, 775];
clad := [61, 47, 119, 47, 61, 83, 83, 47, 97, 47, 83, 83];

47 intervals, 80 triads

# {225/224, 441/440} in 72-et version

# lester

les := [0, 5, 12, 16, 23, 30, 35, 42, 46, 53, 58, 65];
lesd := [5, 7, 4, 7, 7, 5, 7, 4, 7, 5, 7, 7];

46 intervals, 71 triads. Considering this is the least in tune, something
of an also-ran.


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Message: 5128 - Contents - Hide Contents

Date: Wed, 31 Jul 2002 08:38:24

Subject: Re: Four 10-note, 7-limit JI scales

From: Gene W Smith

These scales also work well with the {225/224, 441/440} temperament,
whose mean square optimal values are essentially those of the 72-et. I
give a 72-et version of the first scale below (33 intervals 44 triads);
the third and fourth are modes of the first and second, so the second is
just a mode of the inversion of the first scale. Qm(3) is not knocked off
its perch, but these are a nice suppliment.

! mecaa.scl
! [5, 11, 7, 7, 5, 7, 11, 5, 7, 7]
{225/224, 441/440} tempering of decad, 72-et version
10
!
83.33333333
266.6666667
383.3333333
500.0000000
583.3333333
700.0000000
883.3333333
966.6666667
1083.333333
2/1


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Message: 5129 - Contents - Hide Contents

Date: Wed, 31 Jul 2002 09:01:18

Subject: Re: Four 10-note, 11-limit JI scales

From: Gene W Smith

We can warp in a bit of 11-limit harmony into these scales by means of
the 
{126/125, 441/440} planar temperament. Each scale now has 34 intervals
and 50 triads in the 11-limit (compare to Qm(3), with 35 intervals and 52
triads.)

! secad.scl
! [34, 51, 22, 29, 34, 22, 51, 34, 29, 22]
{126/125, 176/175} tempering of decad, 328-et version
10
!
124.3902439
310.9756098
391.4634146
497.5609756
621.9512195
702.4390244
889.0243902
1013.414634
1119.512195
2/1

! secab.scl
! [22, 29, 34, 51, 22, 34, 29, 22, 51, 34]
{126/125, 176/175} tempering of decab, 328-et version
10
!
80.48780488
186.5853659
310.9756098
497.5609756
578.0487805
702.4390244
808.5365854
889.0243902
1075.609756
2/1

! secac.scl
! [34, 29, 22, 51, 34, 22, 29, 34, 51, 22]
{126/125, 176/175} tempering of decac, 328-et version
10
!
124.3902439
230.4878049
310.9756098
497.5609756
621.9512195
702.4390244
808.5365854
932.9268293
1119.512195
2/1

! secad.scl
! [34, 51, 22, 29, 34, 22, 51, 34, 29, 22]
{126/125, 176/175} tempering of decad, 328-et version
10
!
124.3902439
310.9756098
391.4634146
497.5609756
621.9512195
702.4390244
889.0243902
1013.414634
1119.512195
2/1


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Message: 5130 - Contents - Hide Contents

Date: Thu, 1 Aug 2002 06:11:51

Subject: Another 12-note scale

From: Gene W Smith

Here's a 12-note scale which is comparable to the ones I just did by
tempering Carl's. I took all the JI scales built from (15/14)^3 (16/15)^4
(21/20)^3 (25/24)^2 which consisted of two indentical tetrachords
separated by a 9/8=15/14 21/20. I got two scales and their inversions,
isomorphic by the 21/20 <==> 25/24 transformation. These scales turned
out to be adapted to the {225/224, 385/384} temperament, and on tempering
I ended up with just one scale (modulo modes) and its inversion. I took
this down a fourth to get some dominant harmony, and ended up with this:

1-21/20-9/8-6/5-5/4-21/16-7/5-3/2-8/5-5/3-7/4-28/15

27 (7-limit) intervals, 20 triads

Tempering it, I got the following:

! tetra.scl
! [61, 83, 83, 47, 61, 83, 83, 83, 47, 61, 83, 83]
{225/224, 385/384} tempering of two-tetrachord 12-note scale
! 858-et version of 1-21/20-9/8-6/5-5/4-21/16-7/5-3/2-8/5-5/3-7/4-28/15
12
!
85.31468531
201.3986014
317.4825175
383.2167832
468.5314685
584.6153846
700.6993007
816.7832168
882.5174825
967.8321678
1083.916084
2/1

46 (11 limit) intervals 74 triads

Something for Carl to think about.


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Message: 5131 - Contents - Hide Contents

Date: Sat, 3 Aug 2002 14:25:14

Subject: Optimized 15-note, 7-limit JI scales

From: Gene W Smith

I did a search on these, using (16/15)^4 (21/20)^3 (25/24)^5 (36/35)^3 as
step sizes, and constraining the search by requiring there to be two
complete tetrads a iifth apart. I found two optimal solutions:

opti15a 42 intervals 37 triads 6 tetrads
[1, 21/20, 28/25, 7/6, 6/5, 5/4, 4/3, 7/5, 35/24, 3/2, 8/5, 5/3, 7/4,
28/15, 35/18]

opti15b 42 intervals 37 triads 6 tetrads
[1, 21/20, 35/32, 7/6, 6/5, 5/4, 4/3, 7/5, 35/24, 3/2, 8/5, 5/3, 7/4,
28/15, 35/18]

These are both good candidates for miracle, where they have in the
11-limit
69 intervals and 128 triads. Both tempered and untempered they are
graph-isomorphic without being isomorphic.


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Message: 5132 - Contents - Hide Contents

Date: Sat, 3 Aug 2002 12:30:31

Subject: Prism plus

From: Gene W Smith

I finally hit a homer in the search for 12-note, 7-limit JI scales,
finding two scales closely related to "prism", but better. I searched
scales which contained two tetrachords a fifth apart 
using (15/14)^3 (16/15)^4 (21/20)^3 (25/24)^2 = 2 as scale steps. I found

! pris.scl
! [16/15, 21/20, 25/24, 15/14, 16/15, 21/20, 15/14, 16/15, 25/24, 21/20,
16/15, 15/14]
optimized (15/14)^3 (16/15)^4 (21/20)^3 (25/24)^2 scale
12
!
16/15
28/25
7/6
5/4
4/3
7/5
3/2
8/5
5/3
7/4
28/15
2/1

Then I did another search, which looked for scales containing at least
one 7-limit tetradchord, and found another, graph-isomorphic scale (it
can be seen as the first scale, taken down a fourth, and transformed so
that two of the degrees are changed by 225/224.) "Prism" and similar
scales were looked at during this search, but these two have it beat.
Here is "prisa":

! prisa.scl
! [21/20, 16/15, 15/14, 25/24, 21/20, 16/15, 15/14, 16/15, 21/20, 25/24,
16/15, 15/14]
optimized (15/14)^3 (16/15)^4 (21/20)^3 (25/24)^2 scale
12
!
21/20
28/25
6/5
5/4
21/16
7/5
3/2
8/5
42/25
7/4
28/15
2/1

The statistics are  

prism 30 intervals 24 triads 4 tetrads
pris 30 intervals 25 triads 5 tetrads
prisa 30 intervals 25 triads 5 tetrads

These three scales become the same when tempered by 225/224; in the
{225/224, 385/384} temperament, they have 49 11-limit intervals and 86
triads.


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Message: 5134 - Contents - Hide Contents

Date: Fri, 09 Aug 2002 22:06:40

Subject: Re: find the root of the function

From: wallyesterpaulrus

--- In tuning-math@y..., "nasnas_100" <nasnas_100@y...> wrote:

> hi i am new student i have a problem > find the root of f(x)=x^3+30x-30 > thanks
what does this have to do with tuning?
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Message: 5135 - Contents - Hide Contents

Date: Sat, 10 Aug 2002 12:10:42

Subject: Polynomials and tuning

From: Gene W Smith

On Fri, 09 Aug 2002 22:06:40 -0000 "wallyesterpaulrus"
<perlich@xxx.xxxx.xxx> writes:
> --- In tuning-math@y..., "nasnas_100" <nasnas_100@y...> wrote:
>> hi i am new student i have a problem >> find the root of f(x)=x^3+30x-30 >> thanks
> what does this have to do with tuning?
Nothing; it looks like a homework problem. However, for your consideration I present the following. I can represent, and so study, a scale by the polynomial whose roots are the scale elements. To do this right, I want the octave to be represented by a prime number; that is, I want a map to primes h such that h(2)=p, p a prime. In that way I have no zero divisors in the ring mod p, or in other words I am in a field. Suppose I want to study Blackjack, which is a scale in Miracle. I can't do it mod 72, since that is composite, and 31 and 41 are a little small and may give me extraneous relationships. The 103-et would probably be fine, but instead I choose the map [4447, 7039, 10317, 12477], using the prime 4447 to represent 2. This gives me the rms optimal values for Miracle tuning, with a secor of 432/4447. I now take the polynomial with roots 432i where i ranges from -10 to 10, which I can reduce mod 4447 without loss of information: x^21-61*x^19-1724*x^17-1045*x^15-28*x^13+1971*x^11-1724*x^9-1326*x^7+114 *x^5-846*x^3+1260*x I also define a polynomial whose roots correspond to the twelve 7-limit consonant intervals, obtaining x^12-1314*x^10+1560*x^8-1735*x^6-1921*x^4+1244*x^2+1202 If I take the resultant of the first polynomial with x-n substituted for x with the second polynomial and factor mod 4447, I get (n+1601)^5*(n-1550)^3*(n+254)*(n+1169)^5*(n+432)^10*(n+1855)^7*(n+559) ^6*(n+686)*(n+2033)^5*(n-1169)^5*(n-254)*(n+991)^7*(n-1855)^7*(n-610)*(n- 1982)^4*(n-686)*(n-991)^7*(n+305)^6*(n-737)^6*(n-432)^10*(n+1423)^7*(n+15 50) ^3*n^10*(n-1296)^10*(n-864)^10*(n-1423)^7*(n-1601)^5*(n-305)^6*(n+864)^10 *(n +610)*(n-1042)*(n-127)^6*(n+1042)*(n-1118)^2*(n-1728)^9*(n+1296)^10*(n+11 18) ^2*(n+2160)^8*(n-559)^6*(n+127)^6*(n+1728)^9*(n-2160)^8*(n-178)*(n+1982)^ 4*( n+178)*(n+737)^6*(n-2033)^5 This gives me all the places mod 4447 where we have consonant interval relationships to Blackjack--including those in Blackjack. The multiplicities give the number of consonances. The steps of Blackjack, centered at the unison, are [-2160, -1855, -1728, -1423, -1296, -991, -864, -559, -432, -127, 0, 127, 432, 559, 864, 991, 1296, 1423, 1728, 1855, 2160] and the corresponding multiplicities are [8, 7, 9, 7, 10, 7, 10, 6, 10, 6, 10, 6, 10, 6, 10, 7, 10, 7, 9, 7, 8] If we add this up, we get 170, which is twice the number of 7-limit consonances for Blackjack--twice since we count them twice, once for each scale step.
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Message: 5137 - Contents - Hide Contents

Date: Sat, 10 Aug 2002 23:45:01

Subject: Re: solution of cubic

From: Robert Walker

HI there,

Just sent a post to the freelists.org - all the posts 
sent here get echoed there.

Anyway in summary it has a link to

Quintic equation solution - applet * [with cont.]  (Wayb.)

where I have put up a solution to the quintic that I did
basically for fun and because when I looked for 
on-line pages for solving the quintic I couldn't find any,
so it might be somethign of a gap in the range of Web javascripts
available right now.

Of course you can use it for solving a cubic by setting
the first two coefficients to 0.

That page also has links to a couple of other pages on the web about the
modern solution of the cubic, and its history.

I don't know of any relevance of my page to tuning but did use
solutions of cubic when exploring fibonacci tonescapes
- to find appropriate ratios to use such that if you
go up by one ratio and down by another ratio on the
long and short beats of a fibonacci rhythm, then
in the long term (like hours) you want the pitch
to wander not too far from the original 1/1.

If you choose the numbers right then even after an hour
or so, even with small ratios, you can stay within a 
fraction of a cent of the original 1/1.

If you follow that through you end up with a 
cubic equation to solve.

Robert


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Message: 5138 - Contents - Hide Contents

Date: Sat, 10 Aug 2002 23:50:48

Subject: Re:solution of cubic

From: Robert Walker

HI there,

Sorry getting muddled. When you have two beats
in the pattern you just need to solve a quadratic.
The cubics come in when you look for ratios to use
for fibonacci tonescapes with three beat
fibonacci rhythms.

I expect if one went up to fibonacci patterns
built up using four or more beats you would probably need the quartic
and quintic - that's just a guess as I haven't
worked it out.

Robert


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Message: 5139 - Contents - Hide Contents

Date: Sat, 10 Aug 2002 17:58:22

Subject: Re: find the root of the function

From: Gene W Smith

On Sat, 10 Aug 2002 13:59:06 -0700 "M. Edward Borasky"
<znmeb@xxxxxxx.xxx> writes:
> Well . according to Derive, there are two complex roots and one real > root: > > x = -0.4848069410 - 5.541219478.i ; x = -0.4848069410 + > 5.541219478.i ; x > = 0.9696138820 > > Now that I've given you the answer, your assignment is to look up the > formula (and there is one)
There's more than one, and solving it in radicals (as opposed to Chebyshev radicals, for instance) isn't the neatest.
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Message: 5140 - Contents - Hide Contents

Date: Tue, 13 Aug 2002 18:49:00

Subject: Re: A common notation for JI and ETs

From: gdsecor

Note:  Dave Keenan has kindly agreed to work with me (off-list) on 
the notation project again for a short time to deal with the latest 
modifications that I am proposing.  (Will there ever be an end to 
this?  I think there's light at the end of the tunnel.)  Otherwise, I 
expect that he will continue to take time off from the Tuning List.  
We will be posting our correspondence here to maintain a complete 
record of how the notation is being developed.

I have a long reply to his last message, and I will post this in 
installments.  --GS

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4532]:
> At 01:03 18/06/02 -0000, you wrote:
>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>>>> I would therefore recommend going back to the rational >>>> complementation system and doing the ET's that way as well. >>>
>>> Agreed. Provided we _always_ use rational complements, whether this >>> results in matching half-apotomes or not. >>
>> In other words, you would prefer having this: >> >> 152 (76 ss.): )| |~ /| |\ ~|) /|) /|\ (|) (|\
||~ /|| ||\ ~||) (||~ /||\
>> >> instead of this: >> >> 152 (76 ss.): )| |~ /| |\ /|~ /|) /|\ (|) (|\
||~ /|| ||\ /||~ /||) /||\
>> >> even if it isn't as easy to remember. >
> OK. I think you've got me there. :-) Remember I said I thought we
shouldn't let complements cause us to choose an inferior set of single-shaft symbols, because some people won't use the purely saggital complements. I think we both agree that /|~ is a better choice for 5deg152 than ~|) since it introduces fewer new flags and puts the ET value closer to the rational value. New Rational Complements Part 1 --------------------------------- Now that I've talked you into this, I'm going to have to try to talk you out of it (to some extent) because of something that I have come to realize over these past few weeks. There's nothing like some time off to create a new perspective: I have come back to this as if I were a JI composer new to the notation who is asking the question, "How would I notate a 15-limit tonality diamond?" 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.) 1/1 = C 2/1 = C 3/2 = G 4/3 = F 5/3 = A\! 6/5 = Eb/| or E!!/ 5/4 = E\! 8/5 = Ab/| or A!!/ 7/4 = Bb!) or B!!!) 8/7 = D|) 7/5 = Gb!( or G!!!( 10/7 = F#|( or F|||( 7/6 = Eb!) or E!!!) 12/7 = A|) 9/5 = Bb/| or B!!/ 10/9 = D\! 9/7 = E|) 14/9 = Ab!) or A!!!) 9/8 = D 16/9 = Bb or B\!!/ 11/6 = B(!) 12/11 = D\!/ 11/7 = G#(! or G)||~ 14/11 = Fb(| or F)||~ 11/8 = F/|\ 16/11 = G\!/ 11/9 = E(!) 18/11 = A\!/ 11/10 = D\!~ 20/11 = Bb/|~ or B~!!( 13/7 = B\!~ 14/13 = Db/|~ or D~!!( 13/8 = A(!/ 16/13 = E\!) 13/9 = G(!/ 18/13 = F(|\ 13/10 = F\\! 20/13 = G//| 13/11 = Eb!( or E!!!( 22/13 = A|( 13/12 = D(!/ 24/13 = B\|) 15/8 = B\! 16/15 = Db/| or D!!/ 15/11 = F/|~ 22/15 = G/!~ 15/13 = D//| 26/15 = Bb\\! or B\\!!! 15/14 = C#|( or C|||( 28/15 = Cb!( or C!!!( In determining the notation for all of the 15-odd-limit consonances I found that the symbols of the sagittal notation fall into three groups: 1) those that are very useful for the 15 limit, 2) those that are useful only for primes above 13, and 3) those for which I haven't yet found a use. (Symbols not having rational complements should be in the third category, which is not the case at present.) This is relevant to your very next statement:
> 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. 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) 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. 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 ET's, including 217. (More about this later.)
> But if we look at complements consistent with 494-ET (as all the
rational complements are) the only complement for /|~ is ~||(. So we end up with
> > 152 (76 ss.): )| |~ /| |\ /|~ /|) /|\ (|) (|\ ~||
( /|| ||\ ~||) /||) /||\
> > But this is bad because the flag sequence is different in the two
half-apotomes _and_ ~||( = 10deg152 is inconsistent _and_ too many flag types. So you're right. I don't want to use strict rational complements for this, particularly with its importance in representing 1/3 commas. I'd rather have
>
>> 152 (76 ss.): )| |~ /| |\ /|~ /|) /|\ (|) (|\
||~ /|| ||\ /||~ /||) /||\ I don't follow the part about ~||( = 10deg152 being inconsistent: The 17' comma ~|( is 2deg, and the apotome (15deg) minus the unidecimal diesis (7deg) is (|) = 8deg, so (|) + ~|( = ~||( = 10deg. So I would replace |~, the 23-comma, with ~|(, the 17' comma, which gives: 152 (76 ss.): )| ~|( /| |\ /|~ /|) /|\ (|) (|\ ~||( /|| ||\ /||~ /||) /||\ (RC w/ 14deg AC) This not only uses a symbol ~|( that corresponds to a lower prime symbol for 2deg, but also uses a rational symbol ~||( that has meaning for certain ratios of 11 and 13, as also will /|~. The 14deg symbol /||) is not the rational complement of 1deg )|, but its offset (~1.12 cents) is small enough that it would have qualified as a rational complement (RC) if we had no other choice. I'll call this an alternate complement (AC) -- one that may be used for notating an ET in the absence of a RC consistent in that ET, but which is not used for rational notation. 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. And these are the symbols that should have first priority in the assignment of rational complements. 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. This will be continued, following a short digression about 76-ET.
> I note that 76-ET can also be notated using its native fifth, as
you give (and I agree) below. In the process of looking over what we discussed regarding 76 (in connection with 62 and 69 a bit later in your message #4532), I noticed that it was given above as a subset of 152. I then noticed how bad the 5-limit is in 76 and wondered why it was being considered on its own. I then reviewed our correspondence. In response to a question from Paul about 76-ET, you told him this (in message #4272): << The native best-fifth of 76-ET is not suitable to be used a notational fifth because, among other reasons, it is not 1,3,9- consistent (i.e. its best 4:9 is not obtained by stacking two of its best 2:3s) and I figure folks have a right to expect C:D to be a best 4:9 when commas for primes greater than 9 are used in the notation. So 76-ET will be notated as every second note of 152-ET. >> It gets even worse than this: not only is 3 over 45 percent of a degree false, but 5 deviates even more. Your next mention of 76-ET was in message #4434, in which you treated the divisions of the apotome systematically: 4 steps per apotome ... 69,76: |) ?? (|\ /||\ [13-comma] From that point we had 76 listed both as a subset of 152 and with 69. So after looking at all this, which will it be? (I would prefer it as the subset.) (To be continued.)
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Message: 5141 - Contents - Hide Contents

Date: Tue, 13 Aug 2002 19:19:58

Subject: Re: A common notation for JI and ETs

From: David C Keenan

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 > >Note: Dave Keenan has kindly agreed to work with me (off-list) on the >notation project again for a short time to deal with the latest >modifications that I am proposing. (Will there ever be an end to this? > I think there's light at the end of the tunnel.) Otherwise, I expect >that he will continue to take time off from the Tuning List. We will >be posting our correspondence here to maintain a complete record of how >the notation is being developed.
That's right. I am not reading any lists. Only CCing my replies to George, to tuning-math.
>I have a long reply to his last message, and I will post this in >installments. --GS > >--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4532]:
>> At 01:03 18/06/02 -0000, you wrote:
>>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote: >>> --- In tuning-math@y..., "dkeenanuqnetau" <d.keenan@u...> wrote:
>>>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
>>>>> I would therefore recommend going back to the rational >>>>> complementation system and doing the ET's that way as well. >>>>
>>>> Agreed. Provided we _always_ use rational complements, whether >this
>>>> results in matching half-apotomes or not. >>>
>>> In other words, you would prefer having this: >>> >>> 152 (76 ss.): )| |~ /| |\ ~|) /|) /|\ (|) (|\ ||~
>/|| ||\ ~||) (||~ /||\ >>>
>>> instead of this: >>> >>> 152 (76 ss.): )| |~ /| |\ /|~ /|) /|\ (|) (|\ ||~
>/|| ||\ /||~ /||) /||\ >>>
>>> even if it isn't as easy to remember. >>
>> OK. I think you've got me there. :-) Remember I said I thought we
>shouldn't let complements cause us to choose an inferior set of >single-shaft symbols, because some people won't use the purely saggital >complements. I think we both agree that /|~ is a better choice for >5deg152 than ~|) since it introduces fewer new flags and puts the ET >value closer to the rational value. > >New Rational Complements Part 1 >--------------------------------- > >Now that I've talked you into this, I'm going to have to try to talk >you out of it (to some extent) because of something that I have come to >realize over these past few weeks. There's nothing like some time off >to create a new perspective: I have come back to this as if I were a JI >composer new to the notation who is asking the question, "How would I >notate a 15-limit tonality diamond?"
An excellent question. I think I posed a similar one earlier, but only considering the 11-limit diamond.
>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 5ASCII symbols are being used to represent a single sagittal symbol. How big is the biggest schisma involved?
>> 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.
>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 ET's, 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?
>> But if we look at complements consistent with 494-ET (as all the
>rational complements are) the only complement for /|~ is ~||(. So we >end up with >>
>> 152 (76 ss.): )| |~ /| |\ /|~ /|) /|\ (|) (|\ ~||(
>/|| ||\ ~||) /||) /||\ >>
>> But this is bad because the flag sequence is different in the two
>half-apotomes _and_ ~||( = 10deg152 is inconsistent _and_ too many flag >types. So you're right. I don't want to use strict rational complements >for this, particularly with its importance in representing 1/3 commas. >I'd rather have >>
>>> 152 (76 ss.): )| |~ /| |\ /|~ /|) /|\ (|) (|\ ||~
>/|| ||\ /||~ /||) /||\ > >I don't follow the part about ~||( = 10deg152 being inconsistent: The >17' comma ~|( is 2deg, and the apotome (15deg) minus the unidecimal >diesis (7deg) is (|) = 8deg, so (|) + ~|( = ~||( = 10deg.
My mistake. Sorry.
>So I would replace |~, the 23-comma, with ~|(, the 17' comma,
Well of course I think of |~ as 19'-19 when notating ETs.
>which gives: > >152 (76 ss.): )| ~|( /| |\ /|~ /|) /|\ (|) (|\ ~||( /|| ||\ > /||~ /||) /||\ (RC w/ 14deg AC)
Unfortunately this gives up a desirable property: Monotonicity of flags-per-symbol with scale degree.
>This not only uses a symbol ~|( that corresponds to a lower prime >symbol for 2deg, but also uses a rational symbol ~||( that has meaning >for certain ratios of 11 and 13, as also will /|~. The 14deg symbol >/||) is not the rational complement of 1deg )|, but its offset (~1.12 >cents) is small enough that it would have qualified as a rational >complement (RC) if we had no other choice. I'll call this an alternate >complement (AC) -- one that may be used for notating an ET in the >absence of a RC consistent in that ET, but which is not used for >rational notation. Fair enough. >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 onlyinvolve ratios as uncommon as 5:11 and 7:13?
>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 manyof those 15-limit "consonances" are not consonant at all, and are not evenJust, being indistinguishable from the intervals in their vicinity, exceptif they are a subset of a very large otonality or with the most contrived timbre.
>This will be continued, following a short digression about 76-ET. >
>> I note that 76-ET can also be notated using its native fifth, as you
>give (and I agree) below. > >In the process of looking over what we discussed regarding 76 (in >connection with 62 and 69 a bit later in your message #4532), I noticed >that it was given above as a subset of 152. I then noticed how bad the >5-limit is in 76 and wondered why it was being considered on its own. > >I then reviewed our correspondence. In response to a question from >Paul about 76-ET, you told him this (in message #4272): > ><< The native best-fifth of 76-ET is not suitable to be used a >notational fifth because, among other reasons, it is not >1,3,9-consistent (i.e. its best 4:9 is not obtained by stacking two of >its best 2:3s) and I figure folks have a right to expect C:D to be a >best 4:9 when commas for primes greater than 9 are used in the >notation. So 76-ET will be notated as every second note of 152-ET. >> > >It gets even worse than this: not only is 3 over 45 percent of a degree >false, but 5 deviates even more. > >Your next mention of 76-ET was in message #4434, in which you treated >the divisions of the apotome systematically: > >4 steps per apotome ... >69,76: |) ?? (|\ /||\ [13-comma] >
>> From that point we had 76 listed both as a subset of 152 and with 69.
>So after looking at all this, which will it be? (I would prefer it as >the subset.)
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 Isay give both, but favour the 152-ET subset. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5142 - Contents - Hide Contents

Date: Wed, 14 Aug 2002 18:07:04

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> 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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Message: 5143 - Contents - Hide Contents

Date: Wed, 14 Aug 2002 18:25:04

Subject: Re: A common notation for JI and ETs

From: gdsecor

(This is a continuation of message #4577.)

--- In tuning-math@y..., David C Keenan <d.keenan@u...> wrote [#4532]:
> At 01:03 18/06/02 -0000, you wrote:
>> --- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
New Rational Complements Part 2 --------------------------------- Since I proposed changing a pair of rational complements above, I would like to re-examine the subject of rational complements a bit further. One flag combination that is very useful is 5+5, //|: 25/16 = G#\\! or G~|| 13/10 = F\\! 26/15 = Bb\\! or B\\!!! 20/13 = G//| 15/13 = D//| 17/14 = Eb//| or E~!! and //|| is useful for 17 complements: 17/16 = C#~! or C//|| 17/12 = F#~! or F//|| 32/17 = Cb~| or C//!! 24/17 = Gb~| or G//!! So we see that //| functions not only as the 5+5 comma (6400:6561, ~43.013c) but also as the 13'-5 comma (39:40, ~43.831c) and the 17'+7 comma (448:459, ~41.995c). 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). 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?) 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.) 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. 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. With this set of symbols there are only two intervals (including inversions) in the 15-odd limit that, relative to C=1/1, require symbols outside of the standard 217 set for the rational notation: 11/7 = G#(! or G)||~ 14/11 = Fb(| or F)||~ This uses (| as the 11'-7 comma (45056:45927, ~33.148c), which is already defined as part of the notation. Curiously, these could easily be incorporated into the 217 notation by replacing |\ and /|| with (| with )||~, respectively: 217-mapped JI: |( ~| ~|( /| |) (| /|~ //| /|) /|\ (|) (|\ ~|| ~||( )||~ ||) ||\ /||~ //|| /||) /||\ No 15-odd limit consonances require either |\ or /|| (they are not needed until 19/14 and 38/21 are encountered), so no 15-limit symbols are lost in the process. This completely minimizes the differences between the 217-ET and the rational notation. Rational complementation is maintained, but the matching sequence of symbols is lost (an important consideration with this many symbols), not to mention losing half of the easy-to-remember straight-flag symbols. Also, while the lateral confusability between the straight- flag symbols has been eliminated, that has been replaced by laterally mirrored 14deg and 15deg symbols. So I think it would be best to retain the straight flags in the standard 217 set, 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). 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). (To be continued.)
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Message: 5144 - Contents - Hide Contents

Date: Wed, 14 Aug 2002 19:11:07

Subject: Re: A common notation for JI and ETs

From: David C Keenan

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.
>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.
>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, notthe 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. ...
>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.
>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? Is )| still to be interpreted as the 19 comma and what is to be its complement? Is there a lower prime interpretation of |~ now too? It seems to me that what we are discussing here is unlikely to impact on many ETs below 100. Is that the case? -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5145 - Contents - Hide Contents

Date: Thu, 15 Aug 2002 19:28:14

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., David C Keenan <d.keenan@u...> 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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Message: 5146 - Contents - Hide Contents

Date: Thu, 15 Aug 2002 19:00:22

Subject: Re: A common notation for JI and ETs

From: David C Keenan

At 12:31 15/08/02 -0700, George Secor wrote:
>> 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.
That's a little more strict that what I had in mind, but I guess it's a good idea. I'd be inclined to allow it to represent two 5-commas whether that gives the best 25 or not.
>>> 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.
This seems a little circular. If we did not limit ET notations to using only those symbols used for 15-limit JI, but instead tried to minimise the number of different flags each uses (as we have been until recently), then ~|)may well have wider use than /|~, purely due to the number of ETs it is used in. So I don't buy this one.
>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.
Yes, I agree that is the whole point of our "common notation". However I'm not convinced that there will be many times when somone uses an approximate11:15 or 13:14 _as_ an approximate just interval when the lower note is a natural (or has only # or b). But in the case of a 5:11 I guess it's more likely. So I find this reason to be marginally valid.
>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.
On examing these in more detail I find that I don't understand at all why you chose /|~ as the appropriate symbol for the 11'-5 comma, 44:45 (and the 13'-7 comma). (|( seems the obvious choice to me, since (| is the 11'-7 comma and |( is the 7-5 comma and (| + |( = (|( . (11'-7)+(7-5) = 11'-5. and (11'-7)+(13'-11')=(13'-7). If (|( is the symbol for the 11'-5 comma (or we could more usefully call itthe 11/5 comma) then you don't need to change any rational complements from what we had (the 494-ET-consistent ones) and what's more we don't need tointroduce any more flags into 217-ET when (|( is used for 7 steps.
>> 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.)
You're only looking at the primes themselves. What about the ratios betweenthem. 217-ET has a 2.8 cent error in its 7:11 whereas 282-ET never gets worse than that 2.0 cents in the 1:13.
>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.
Reason enough to reject 282-ET as what? Reject it as a good way of having afully notatable closed system that approximates 29-limit JI? I seriously disagree. It just means that we should use (| and |) with their non-13 meanings in 282-ET.
>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
All this means is that maybe we should consider making our rational complements consistent with 282-ET rather than 217-ET.
>What makes 217 so useful is that *everything* is consistent to the 19 >limit, and, except for 23, to the 29 limit.
I don't know what you mean by *everything* here. Isn't 282-ET consistent tothe 29-limit with no exceptions?
>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!
I still fail to see why 217 is better than 282, except that various choiceswe have made along the way, regarding the symbols, have been biased toward217.
>>> 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
No. Although that's interesting too.
>But now I think you meant listing the ratios like this: > >Sagittal Notation for 15-limit JI >--------------------------------- ...
Yes that was it, but now I realise there are only 6 that are independent and that we haven't already agreed on. Here they are in oder of decreasing importance: 1/1 = C 7/5 = Gb!( or G!!!( 11/5 = D\!~ 11/7 = G#(! or G)||~ 13/5 = F\\! 13/7 = B\!~ 13/11 = Eb!( or E!!!( But I think they should be: 1/1 = C 7/5 = Gb!( or G!!!( 11/5 = D(!( 11/7 = G#(! or G)||~ 13/5 = F\\! 13/7 = B(!( 13/11 = Eb!( or E!!!(
>> 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.
I mean: What's the smallest one we've agreed on that uses |(, where the 7-5comma interpretation of it would be a different number of steps from what we've used it for.
>> 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.
No. Going via complements isn't what I had in mind. Does )| want to be usedas a comma for any of 17/5, 17/7, 17/11, 17/13?
>> 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.
OK. But this is not so, if we adopt (|( as the 7/5-comma symbol. -- Dave Keenan Brisbane, Australia Dave Keenan's Home Page * [with cont.] (Wayb.)
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Message: 5147 - Contents - Hide Contents

Date: Fri, 16 Aug 2002 18:49:50

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- David C Keenan <d.keenan@xx.xxx.xx> wrote:
> At 12:31 15/08/02 -0700, George Secor wrote:
>>> 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. >
> That's a little more strict that what I had in mind, but I guess it's > a good idea. I'd be inclined to allow it to represent two 5-commas > whether that gives the best 25 or not.
In working out a spreadsheet to automatically assign the symbols for ETs, one of the criteria I am using is to select ones that eliminate (or at least minimize) the inconsistencies. This can get not only complicated, but tricky.
>>>> 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. >
> This seems a little circular. If we did not limit ET notations to > using only those symbols used for 15-limit JI, but instead tried to > minimise the number of different flags each uses (as we have been > until recently), then ~|) may well have wider use than /|~, purely > due to the number of ETs it is used in. So I don't buy this one. >
>> 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. >
> Yes, I agree that is the whole point of our "common notation". > However I'm not convinced that there will be many times when somone > uses an approximate 11:15 or 13:14 _as_ an approximate just interval > when the lower note is a natural (or has only # or b). But in the > case of a 5:11 I guess it's more likely. So I find this reason to be > marginally valid. >
>> 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. >
> On examing these in more detail I find that I don't understand at all > why you chose /|~ as the appropriate symbol for the 11'-5 comma, > 44:45 (and the 13'-7 comma). (|( seems the obvious choice to me, > since (| is the 11'-7 comma and |( is the 7-5 comma and (| + |( = (| (. > (11'-7)+(7-5) = 11'-5. and (11'-7)+(13'-11')=(13'-7).
Oops, you're right! I've been using (| as the 13'-(11-5) comma all along in computing these ratios, and it looks like I'm going to have to redo a few things on account of our recent lower-prime symbol definitions. Glad you caught this! Of course, the 13'-7 comma is 1664:1701, about 0.833 cents smaller. So (|( will definitely have to be among the 217 standard symbols, and it's back to the drawing board! (Really, I'm very delighted that you found this, because it's going to make things a lot easier.) So it looks like this will be the 217 standard set: 217: |( ~| ~|( /| |) |\ (|( //| /|) /|\ (|) (|\ ~|| ~|| ( /|| ||) ||\ (||( //|| /||) /||\ (new RCs)
> If (|( is the symbol for the 11'-5 comma (or we could more usefully > call it the 11/5 comma) then you don't need to change any rational > complements from what we had (the 494-ET-consistent ones) and what's > more we don't need to introduce any more flags into 217-ET when (|( > is used for 7 steps. >
>>> 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.) >
> You're only looking at the primes themselves. What about the ratios > between them. 217-ET has a 2.8 cent error in its 7:11 whereas 282-ET > never gets worse than that 2.0 cents in the 1:13. >
>> 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. >
> Reason enough to reject 282-ET as what? Reject it as a good way of > having a fully notatable closed system that approximates 29-limit JI? > I seriously disagree. It just means that we should use (| and |) with > their non-13 meanings in 282-ET.
I guess I didn't get my point across. I want to be able to use a large-numbered ET (217 or 282 or whatever) to notate *JI* when there are no suitable rational symbols that will do the job. If (| or |) don't have 13 meanings in 282, then there cannot be a good transition between the rational notation and the large-ET notation -- symbols would have to be converted from one to the other should a JI composition suddenly require 282-ET symbols. This problem is minimized with 217, because even the non-standard symbols such as )| and (| can be kept, because they are all the correct number of degrees.
>> 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 >
> All this means is that maybe we should consider making our rational > complements consistent with 282-ET rather than 217-ET. >
>> What makes 217 so useful is that *everything* is consistent to the 19 >> limit, and, except for 23, to the 29 limit. >
> I don't know what you mean by *everything* here. Isn't 282-ET > consistent to the 29-limit with no exceptions?
It isn't consistent with the schismas that are essential to the rational notation: 1) The 5 comma /| (5deg) plus the 7 comma |) (6deg) doesn't equal the 13 comma /|) (12deg); this is the 4095:4096 schisma, ~0.423c. So you can't notate ratios of 7 that are consistent with ratios of 13 in 282. 2) The 17'-17 comma (2deg) doesn't equal the 7-5 (1deg), or put another way, |) <> /|(; this is the 163840:163863 schisma, ~0.243c. So you can't notate ratios of 17 that are consistent with ratios of 7 and 13 in 282. Or should we discard these and start over -- I think I would then be entitled to say that you have either a 288-bias or an anti-217 bias.
>> 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! >
> I still fail to see why 217 is better than 282, except that various > choices we have made along the way, regarding the symbols, have been > biased toward 217.
My latest solution for the 23' comma is actually biased more toward low error and 494 than it is toward 217, as you'll see in the continuation of my reply to your message #4543.
> ...
>>> 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. >
> I mean: What's the smallest one we've agreed on that uses |(, where > the 7-5 comma interpretation of it would be a different number of > steps from what we've used it for.
Our latest agreement has been on mostly ETs below 100, and I don't think any of those even used |(. The larger-numbered ones were still subject to review at the time you took your break, so they are still open to review. I said that 193 would be affected, but the 19 comma ) | can be used instead of (| for 1deg, so that's no problem.
>>> 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. >
> No. Going via complements isn't what I had in mind. Does )| want to > be used as a comma for any of 17/5, 17/7, 17/11, 17/13? No.
>>> 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. >
> OK. But this is not so, if we adopt (|( as the 7/5-comma symbol.
True (except that you meant the 11/5 comma, but I would prefer calling it the 11'-5 comma for now). I'm going to have to take some time to figure out how everything works out with this change (even if it is a change back for me), since I've been using /|~ for the past couple of weeks. But there is no question that we should use (|(. --George
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Message: 5148 - Contents - Hide Contents

Date: Fri, 16 Aug 2002 18:52:50

Subject: Re: A common notation for JI and ETs

From: gdsecor

--- In tuning-math@y..., "gdsecor" <gdsecor@y...> wrote:
> --- David C Keenan <d.keenan@u...> wrote:
>> At 12:31 15/08/02 -0700, George Secor wrote:
>>>> 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. >>
>> That's a little more strict that what I had in mind, but I guess > it's
>> a good idea. I'd be inclined to allow it to represent two 5-commas >> whether that gives the best 25 or not. >
> In working out a spreadsheet to automatically assign the symbols for > ETs, one of the criteria I am using is to select ones that eliminate > (or at least minimize) the inconsistencies. This can get not only > complicated, but tricky. >
>>>>> 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. >>
>> This seems a little circular. If we did not limit ET notations to >> using only those symbols used for 15-limit JI, but instead tried to >> minimise the number of different flags each uses (as we have been >> until recently), then ~|) may well have wider use than /|~, purely >> due to the number of ETs it is used in. So I don't buy this one. >>
>>> 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. >>
>> Yes, I agree that is the whole point of our "common notation". >> However I'm not convinced that there will be many times when somone >> uses an approximate 11:15 or 13:14 _as_ an approximate just interval >> when the lower note is a natural (or has only # or b). But in the >> case of a 5:11 I guess it's more likely. So I find this reason to be >> marginally valid. >>
>>> 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. >>
>> On examing these in more detail I find that I don't understand at > all
>> why you chose /|~ as the appropriate symbol for the 11'-5 comma, >> 44:45 (and the 13'-7 comma). (|( seems the obvious choice to me, >> since (| is the 11'-7 comma and |( is the 7-5 comma and (| + |( = (| > (.
>> (11'-7)+(7-5) = 11'-5. and (11'-7)+(13'-11')=(13'-7). >
> Oops, you're right! I've been using (| as the 13'-(11-5) comma all > along in computing these ratios, and it looks like I'm going to have > to redo a few things on account of our recent lower-prime symbol > definitions. Glad you caught this! Of course, the 13'-7 comma is > 1664:1701, about 0.833 cents smaller. So (|( will definitely have to > be among the 217 standard symbols, and it's back to the drawing > board! (Really, I'm very delighted that you found this, because it's > going to make things a lot easier.) > > So it looks like this will be the 217 standard set: > > 217: |( ~| ~|( /| |) |\ (|( //| /|) /|\ (|) (|\ ~|| ~|| > ( /|| ||) ||\ (||( //|| /||) /||\ (new RCs) >
>> If (|( is the symbol for the 11'-5 comma (or we could more usefully >> call it the 11/5 comma) then you don't need to change any rational >> complements from what we had (the 494-ET-consistent ones) and what's >> more we don't need to introduce any more flags into 217-ET when (| ( >> is used for 7 steps. >>
>>>> 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.) >>
>> You're only looking at the primes themselves. What about the ratios >> between them. 217-ET has a 2.8 cent error in its 7:11 whereas 282- ET >> never gets worse than that 2.0 cents in the 1:13. >>
>>> 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. >>
>> Reason enough to reject 282-ET as what? Reject it as a good way of >> having a fully notatable closed system that approximates 29-limit > JI?
>> I seriously disagree. It just means that we should use (| and |) > with
>> their non-13 meanings in 282-ET. >
> I guess I didn't get my point across. I want to be able to use a > large-numbered ET (217 or 282 or whatever) to notate *JI* when there > are no suitable rational symbols that will do the job. If (| or |) > don't have 13 meanings in 282, then there cannot be a good transition > between the rational notation and the large-ET notation -- symbols > would have to be converted from one to the other should a JI > composition suddenly require 282-ET symbols. This problem is > minimized with 217, because even the non-standard symbols such as ) | > and (| can be kept, because they are all the correct number of > degrees. >
>>> 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 >>
>> All this means is that maybe we should consider making our rational >> complements consistent with 282-ET rather than 217-ET. >>
>>> What makes 217 so useful is that *everything* is consistent to the > 19
>>> limit, and, except for 23, to the 29 limit. >>
>> I don't know what you mean by *everything* here. Isn't 282-ET >> consistent to the 29-limit with no exceptions? >
> It isn't consistent with the schismas that are essential to the > rational notation: > > 1) The 5 comma /| (5deg) plus the 7 comma |) (6deg) doesn't equal the > 13 comma /|) (12deg); this is the 4095:4096 schisma, ~0.423c. So you > can't notate ratios of 7 that are consistent with ratios of 13 in 282. > > 2) The 17'-17 comma (2deg) doesn't equal the 7-5 (1deg), or put > another way, |) <> /|(; this is the 163840:163863 schisma, ~0.243c. > So you can't notate ratios of 17 that are consistent with ratios of 7 > and 13 in 282. > > Or should we discard these and start over -- I think I would then be > entitled to say that you have either a 288-bias or an anti-217 bias. >
>>> 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! >>
>> I still fail to see why 217 is better than 282, except that various >> choices we have made along the way, regarding the symbols, have been >> biased toward 217. >
> My latest solution for the 23' comma is actually biased more toward > low error and 494 than it is toward 217, as you'll see in the > continuation of my reply to your message #4543. > >> ...
>>>> 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. >>
>> I mean: What's the smallest one we've agreed on that uses |(, where >> the 7-5 comma interpretation of it would be a different number of >> steps from what we've used it for. >
> Our latest agreement has been on mostly ETs below 100, and I don't > think any of those even used |(. The larger-numbered ones were still > subject to review at the time you took your break, so they are still > open to review. I said that 193 would be affected, but the 19 comma ) > | can be used instead of (| for 1deg, so that's no problem. >
>>>> 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. >>
>> No. Going via complements isn't what I had in mind. Does )| want to >> be used as a comma for any of 17/5, 17/7, 17/11, 17/13? > > No. >
>>>> 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. >>
>> OK. But this is not so, if we adopt (|( as the 7/5-comma symbol. >
> True (except that you meant the 11/5 comma, but I would prefer > calling it the 11'-5 comma for now). > > I'm going to have to take some time to figure out how everything > works out with this change (even if it is a change back for me), > since I've been using /|~ for the past couple of weeks. But there is > no question that we should use (|(. > > --George
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Message: 5149 - Contents - Hide Contents

Date: Mon, 19 Aug 2002 15:12:19

Subject: Re: A common notation for JI and ETs

From: gdsecor

(This is a continuation of my message #4580, which is in reply to 
Dave Keenan's message #4543.)

New Rational Complements  Part 3
---------------------------------

You previously mentioned that all of the rational complements are 
consistent with 494-ET (as they are also with 217-ET).  I would like 
to define another pair of supplementary rational complements; we 
didn't need these before, but they just might be useful when we're 
doing some of the more obscure ETs.  They're consistent in both 217 
and 494, and the offset is 0.49 cents.  They are:

~|~ <--> /||( and
/|( <--> ~||~

There are at least a couple of ratios that these can be used to 
notate:

19/10 = Cb~|~ or C\!(
19/15 = Fb~|~ or F\!(

Also, we might want to allow both /|( and ~|~ as their own alternate 
complements in certain instances:

/|( <--> /||(
~|~ <--> ~||~

This is just in case we need them.  I would really not want to use 
these unless it were a last resort.  (After all, I want to keep the 
number of symbols to a minimum.)

New Rational Complements  Part 4
---------------------------------

Now for what may be the most controversial issue -- actually, at the 
last minute I came up with a very non-controversial solution to the 
whole thing (almost a no-brainer), but I'll leave what I had here; 
just don't reply to any of it until you get to the end -- I would 
like to propose a definition of yet another supplementary pair of 
rational complements:

)|( <--> ~||\ and
~|\ <--> )||(

Both of these are symbols that formerly lacked rational complements.  
This is being done so that ~|\, which I am now proposing to be the 
23' comma instead of (|(, may have a rational complement.

The reason that we did not previously use ~|\ as the 23' comma is 
that it lacked a rational complement.  Using ~|\ for this purpose has 
the advantage of making the 23' comma consistent in the majority of 
the best large-numbered ETs, including 152, 171, 217, 224, 270, 311, 
494 (yes, 494 too!!!), and 612, *none* of which use (|( consistently 
as the 23' comma.  (This is one more thing that would make a 
transition between rational notation and 217 notation for JI as easy 
and consistent -- seamless might be a good word -- as possible.)

Another advantage relates to the Reinhard property:  The accuracy for 
(|(, 1441792:1474767, ~39.149c, as the 23' comma, 16384:16767, 
~40.004c, is contingent on the definition of (| as the 13'-(11-5) 
comma (715:729) or as the 29 comma (256:261).  But if (| is defined 
as the 11'-7 comma (45056:45927), then the schisma is 2023:2024, 
~0.856 cents, which is larger than what we have with ~|\, 4352:4455, 
~40.496c, for a schisma of 3519:3520, ~0.492c.  Using ~|\ makes the 
schisma independent of the size of (|.

There are a couple of possible objections to this:

1) The rational complementation offset is ~3.40 cents, which is 
relatively large.  (This would apply only to the single-symbol 
notation.)  I don't think this is much of a problem, because the 
complement symbols are *defined* as rational intervals, not as the 
sum of their component stems and flags.  We wanted to keep the 
offsets low in order to minimize the inconsistencies, but consider 
the alternative: when we had (|( as the 23' comma we had an 
inconsistency for the symbol itself in both 217 and 494; this new 
proposal eliminates that.

2) The rational complement being proposed is consistent in 217, but 
not in 494.  I checked consistency for a number of the better ETs in 
this general neighborhood; most of those under 300 are consistent, 
and all of those above 300 are inconsistent, so it's definitely 
related to the offset.  (Again, this would apply only to the single-
symbol notation, and the inconsistency occurs mostly in systems that 
we are not even going to notate.)

Is it all that important to have all of the rational complements 
consistent with 494?  If it is, then I just got an idea for what may 
be an even better solution, one that you suggested, but with a twist:

<<  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.  >>

You were intending ~)|| to be the complement of (|(, which has the 
following consequences:

1) The complement has an offset of 1.59c with xL as the 13'-(11-5) 
comma, which increases to 2.02 cents if you make xL the 11'-7 comma.

2) The complement is inconsistent in 494, but consistent in 217.

3) And as I said above, the 23' comma itself is inconsistent in both 
217 and 494.

But if we were to make ~)|| the rational complement of ~|\, then:

1) The offset would be 0.67c, independent of the xL flag.

2) The complement would be consistent in 494, but inconsistent in 217.

3) And as I said above, the 23' comma itself would be consistent in 
both 217 and 494.

As for the inconsistency of  the complement in 217, the ~)|| symbol 
could either be replaced with the standard ~|| symbol or else with )||
( to specially designate the 23' complement.  Thus only one obscure 
complementary symbol would have to be changed in going from the 
strict rational to the 217 quasi-rational version.

The foregoing was written before you pointed out that (|( is the true 
11'-5 and 13'-7 comma in your latest message.  In light of this, I 
would still assign ~|\ as the 23' comma, while making (|( a standard 
symbol with rational complement ~||(, thereby eliminating /|~ from 
the picture.  (I was also using /|~ for 17/11 as Ab\!~ or A\!!!~, but 
I'll see how well (|( works later.)  One thing I am very happy about 
is that the lateral confusability between /|~ and ~|\ is eliminated 
if one of those two symbols is eliminated.

So what do you think?

--George


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