4296

Joe has a table for 4296 on one of his new web pages, and takes note of the fact that it is absurdly well in tune in the 5-limit. In fact, if you go out to 200000 it turns out to have the best log-flat badness score:

1 .7369655945
2 .7439736471
3 .4245472985
4 .6797000080
5 .8728704449
7 .6706891205
12 .5418300617
15 .8735997285
19 .5083949041
31 .8578063580
34 .6488389972
53 .4527427539
65 .7839445968
118 .4134357960
171 .6499654469
289 .9207676000
441 .6622791000
559 .9976240155
612 .5675982650
730 .6113208564
1171 .7597497149
1783 .5008376597
2513 .5396476355
4296 .2748910261
6809 .7979361504
8592 .7775092347
16572 .9822272987
20868 .6543929561
25164 .5742465687
52841 .8091483956
73709 .8117671040
78005 .3351525026
151714 .7011086648
156010 .9479544293

Joe credits it to Mark Jones, but it would be interesting to know if it is much older than that--it could have been discovered by hand in the same way 612 and 730 were. I found it about 25 years back cranking a poor, abused TI58 programmable calculator for days at a time doing searchs, where I got up as far as 20868 in the 5-limit, but this brute force method is not really needed in the case of the
5-limit, where various algorithms would turn it up.

The MT reduced basis for 4296 is <2^-90 3^-15 5^49, 2^71 3^-99 5^37>.
The first comma is the smallest one on my list of best 5-limit temperaments, and gvies us the map [[0, 49, 15], [1,-6,0]]. This divides the 5 into 15 parts, and if we tempered 71 or 84 notes by it, we would get a lot of essentially just ratios. If Mark has no objection, perhaps the "Jones" would be a good name for this temperament; the Jones generator being 665/4296, slightly short of satanic. If we take the Jones comma and wedge it with
2^161 3^(-84) 5^(-12) we get h4296 in the 5-limit; this comma I also took note of and it has been mentioned on the main tuning list--by whom initially, I don't know. We can regard 4296 as the intersection of these two absurdly accurate temperaments on the charts of Paul and Joe--if 71 appears on the chart, then the line from 71 to 84, passing through the origin, would be the Jones line, by the way.

So far as performance goes, I wonder if it even makes sense in physical terms to claim to have done this. 4296 gives a fifth which is
.0003064 cents sharp, and a third which is .0008647 cents flat. Obviously we can't hear the difference between this and just intonation, but can we even accomplish it?

If the answer is yes, we can always go on and ask the same question of
78005, whose MT reduced basis, in case anyone cares, is
<2^-573 3^237 5^85, 2^140 3^-374 5^195>

On 2/20/02 2:02 AM, "genewardsmith" <genewardsmith@juno.com> wrote:

> Joe credits it to Mark Jones, but it would be interesting to know if it is
> much older than that--it could have been discovered by hand in the same way
> 612 and 730 were. I found it about 25 years back cranking a poor, abused TI58
> programmable calculator for days at a time doing searchs, where I got up as
> far as 20868 in the 5-limit, but this brute force method is not really needed
> in the case of the 5-limit, where various algorithms would turn it up.
>

Nice to see someone mentioning 20868. That was the highest I ever played
with synergies without actually trying to listen to the temperament since at
the time, 12 years ago, I couldn't create sounds that accurately. I've
spent a lot more than a moment's thought on a lot of 4- and 5-digit
temperaments, but on the list I submitted to Monzo, I only listed the ones
I've really done anything functional with.

What I said was 4296 is the lowest temperament I've found useful for quickly
measuring 5th limit intervals without needing a calculator. That is, in
comparing the folding of intervals with the temperaments I'm trying to work
with.

The octave, 4296 is sort of burned into the brain.

The fifth is 2513/4296. Which is coincidentally the value of another
temperament that shows up in 5th limit convergences. 2513 is 1171 + 1342,
which I've mentioned I'd used as my highest test years ago in testing
temperaments, so on the road, 2513 is easy enough to remember.

(Also the fourth, 1783 would come up a lot, obviously if 2513 and 4296 are
accurate enough to show up in algorithms, then their difference would.)

The major third is 1383. Um okay so I don't know how I manage but well, I
remember that one. Maybe because it looks like 1783.

The syntonic comma is 77/4296.
The pythagorean comma is 84/4296.
That's how I remember the proportion of those two as well.

And from that it's easy to see the schisma is 7/4296.

Anything else, should be easy to figure out.

See, the whole idea of doing it in cents screwed me up. Because eventually
you have to work in a couple decimal points places and even then if you add
or subtract or multiply things too much you still might be a number off. If
you have a nice whole number where everything is accurate enough, well, you
can use it with no worries.

Anyway.

If you run the Brun algorithm with ONLY the fourth or fifth, 4296 shows up
as a 3rd limit temperament. But the fact that it also shows up in so many
5th limit algorithms with so many different shaped convergence webs, I found
it to be extremely ductile.

I ran the Brun-o-matic for about three days years ago doing up to I think
13th limit convergences... Printed out a few dozen pages with the
algorithms in 4 point text. Studied them for quite awhile just to get an
idea of the harmonic makeup of high temperaments. I retained a bit. Not
much in comparison. But after swimming in a place where 10-digit
temperaments became common knowledge, well, back into the finite land,
things seem a lot more easy to grasp.

I'll say it again. NICE to see someone mention 20868!!!

Marc

--- In tuning-math@y..., "Orphon Soul, Inc." <tuning@o...> wrote:

> The octave, 4296 is sort of burned into the brain.

The same thing happened to me with 612; I thought about using 4296 among other possibilities as a system for representing intervals but it seemed like overkill, and too 5-limit.

> (Also the fourth, 1783 would come up a lot, obviously if 2513 and 4296 are
> accurate enough to show up in algorithms, then their difference would.)

This sort of thing seems to happen a lot for some reason.

> See, the whole idea of doing it in cents screwed me up. Because eventually
> you have to work in a couple decimal points places and even then if you add
> or subtract or multiply things too much you still might be a number off.

Right; that's why I switched to 612. Now that I have to deal with other people I switched back to cents again, though.

> I ran the Brun-o-matic for about three days years ago doing up to I think
> 13th limit convergences...

It's possible to miss things using such algorithms; brute force is safer, actually.

> I'll say it again. NICE to see someone mention 20868!!!

Hey, and what about 78005? :)

On 2/21/02 3:24 AM, "genewardsmith" <genewardsmith@juno.com> wrote:

> --- In tuning-math@y..., "Orphon Soul, Inc." <tuning@o...> wrote:
>
>> The octave, 4296 is sort of burned into the brain.
>
> The same thing happened to me with 612; I thought about using 4296 among other
> possibilities as a system for representing intervals but it seemed like
> overkill, and too 5-limit.
>

Ehh. Yeah I can see that. Soundwise, it's easy enough to tell the
difference between 559 and 612. 559 seems like some kind of great unity
until you hear the same thing in 612 when it no longer seems you have to
hold onto the sound for it to be a part of you. And then it gets a little
beyond you, like I said it took an HOUR of listening to 1171 and 1342 to
tell them apart.

>> (Also the fourth, 1783 would come up a lot, obviously if 2513 and 4296 are
>> accurate enough to show up in algorithms, then their difference would.)
>
> This sort of thing seems to happen a lot for some reason.
>

There's all sorts of coincedences aren't there.

>> See, the whole idea of doing it in cents screwed me up. Because eventually
>> you have to work in a couple decimal points places and even then if you add
>> or subtract or multiply things too much you still might be a number off.
>
> Right; that's why I switched to 612. Now that I have to deal with other people
> I switched back to cents again, though.
>

Popular demand has brought me there as well. But you know what, I got so
fed up one day I mapped out the scale tree in cents up to the third level,
which included 12, 17, 19, 22, 26, 27, 29, 31, 33, 39, 41, 43, 45, 46 and
50, and you know what, outside of a little congestion midoctave, and outside
of the pythagorean minor third in 41 being identical in cents to a note in
45 (because of the closeness to 135), EVERY note in those 15 temperaments
had its own unique ID in 1200. And I realized cents isn't so bad a measure
for practical use. If you have to.

I even mapped them to 768 since I'd had a TX81z but wasn't sure if it could
handle tunings with that low a resolution, and even that wasn't so bad.

>> I ran the Brun-o-matic for about three days years ago doing up to I think
>> 13th limit convergences...
>
> It's possible to miss things using such algorithms; brute force is safer,
> actually.
>

I HAVE to start making boards again.

>> I'll say it again. NICE to see someone mention 20868!!!
>
> Hey, and what about 78005? :)
>

Ahh, my brother from another mother! Yes of course. SOOBL on a calculator.
I thought other than almost looking like "school", it's an acronym of
"bloos" which is a fairly illiterate way of spelling "blues", which in the
day wasn't exactly played by schooled people. But then I thought yeah blues
played in 78 thousand notes an octave. That would be umm interesting.

Ever notice 78005 is 15601 times 5? BAAAAAA HAAA!!!

Marc

On 2/20/02 2:02 AM, "genewardsmith" <genewardsmith@juno.com> wrote:

> The first comma is the smallest one on my list of best 5-limit temperaments,
> and gvies us the map [[0, 49, 15], [1,-6,0]]. This divides the 5 into 15
> parts, and if we tempered 71 or 84 notes by it, we would get a lot of
> essentially just ratios. If Mark has no objection, perhaps the "Jones" would
> be a good name for this temperament; the Jones generator being 665/4296,
> slightly short of satanic. If we take the Jones comma and wedge it with ...

Err I didn't see this part before.

(Wrinkling eyebrows, smacking stale aftertaste...)

I umm... Naming a TEMPERAMENT after me? Jeesh. Which one are you saying?
665 or 4296?

Oh actually... HAAAAAAAA

http://www.ixpres.com/interval/dict/marcdefs.htm

...look at the term "Jones" (which I'd forgotten about, it only came up on
the phone with Fred the other night)

Actually if you're talking about 4296 we occasionally called it Silly Putty
because of the way it stretched over different convergence webs.

Nah, man, I've got plenty of geometric and algebraic temperaments I'd much
rather have named after me.

Oh I'll get him back now. BAAAAA HAAAA HAA... Name 4296 "Fred". He might
well have been the one to notice it was all over the place anyway.

I'm kidding either way. Of course I object. :-P

Marc

--- In tuning-math@y..., "Orphon Soul, Inc." <tuning@o...> wrote:

> If you run the Brun algorithm with ONLY the fourth or fifth, 4296 shows up
> as a 3rd limit temperament. But the fact that it also shows up in so many
> 5th limit algorithms with so many different shaped convergence webs, I found
> it to be extremely ductile.

It is a semiconvergent for *both* log2(3) and log2(5) (reducing 9975/4296 to 3325/1432 in the case of log2(5)). This is a rather amazing property, and it would be interesing to know what else, if anything, shares it.

On 2/21/02 5:40 AM, "genewardsmith" <genewardsmith@juno.com> wrote:

> It is a semiconvergent for *both* log2(3) and log2(5) (reducing 9975/4296 to
> 3325/1432 in the case of log2(5)). This is a rather amazing property, and it
> would be interesing to know what else, if anything, shares it.

Semi WHO? If you mean what I think you mean, then yeah 12276.

I let it run a little longer. Also 2588183. Nothing in between.

--- In tuning-math@y..., "Orphon Soul, Inc." <tuning@o...> wrote:
> On 2/20/02 2:02 AM, "genewardsmith" <genewardsmith@j...> wrote:
>
> > The first comma is the smallest one on my list of best 5-limit temperaments,
> > and gvies us the map [[0, 49, 15], [1,-6,0]]. This divides the 5 into 15
> > parts, and if we tempered 71 or 84 notes by it, we would get a lot of
> > essentially just ratios. If Mark has no objection, perhaps the "Jones" would
> > be a good name for this temperament; the Jones generator being 665/4296,
> > slightly short of satanic. If we take the Jones comma and wedge it with ...
>
> Err I didn't see this part before.
>
> (Wrinkling eyebrows, smacking stale aftertaste...)
>
> I umm... Naming a TEMPERAMENT after me? Jeesh. Which one are you saying?
> 665 or 4296?

neither. gene is talking about the _linear_ temperament, not _equal_ temperament, whose _generator_ is about 665/4296 of an octave, but not exactly -- its tuning can be optimized in various ways, so that it will be (inaudibly) different from 4296-equal. kind of like the optimal meantone generator is about 29/50 of an octave, but not exactly . . .

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "Orphon Soul, Inc." <tuning@o...> wrote:
>
> > If you run the Brun algorithm with ONLY the fourth or fifth, 4296 shows up
> > as a 3rd limit temperament. But the fact that it also shows up in so many
> > 5th limit algorithms with so many different shaped convergence webs, I found
> > it to be extremely ductile.
>
> It is a semiconvergent for *both* log2(3) and log2(5) (reducing 9975/4296 to 3325/1432 in the case of log2(5)).

don't forget to try log2(5/3).

>This is a rather amazing property, and it would be interesing to know what >else, if anything, shares it.

if you're allowed to reduce like this, 12 does, because 28/12 = 7/3.

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:

> if you're allowed to reduce like this, 12 does, because 28/12 = 7/3.

Yes, this occurred to me after I posted. It would be interesting to know if anything is a semiconvergent for 3/2, 5/3, and 5/4, but these should be finite in number, and if you don't get one fairly quickly you will probably not get one at all.