Tuning-Math Digests messages 5126 - 5150

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Message: 5126

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

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

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

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

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

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

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: 5135

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

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 *

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

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

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: 5141

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 *


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Message: 5144

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 *


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Message: 5146

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 *


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