Re: [tuning-math] 55-tET

> From: monz <joemonz@yahoo.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Tuesday, December 18, 2001 6:48 PM
> Subject: [tuning-math] 55-tET (was: Re: inverse of matrix --> for what?)
>
>
> What I had in mind was that there should be a pair of
> unison-vectors which defines the set of acoustically implied
> ratios which I put on my lattice at
> <http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm>?
> ... assuming, of course, that in the places where two
> ratios are implied equally well/badly, only one can be chosen.
>
> I find that if I continue my diagram, the unison-vector that
> "works" together with the syntonic comma (-4 4 -1 ) to close
> the system at 55 tones, is the (-51 19 9 ). The 8ve-invariant
> tuning of the 55th quasi-meantone pitch would be 3^19 * 5^(55/6),
> which is ~10.38405963 cents higher than the starting pitch, and
> the ratio it implies most closely is 3^19 * 5^9.

Oops... my bad. Two errors here.

That should say "The 8ve-invariant tuning of the 55th
quasi-meantone *generator*...", calling the starting pitch
the zero-th generator. And the tuning itself is wrong:
it should be 3^(55/3) * 5^(55/6).

(55/3 = 18 & 1/3, and 55/6 = 9 & 1/6. A simple foul-up:
the 54 generator is exactly the ratio 3^18 * 5^9; for the
next one I accidentally added 1 instead of 1/3 to 18.)

The next "closure" size for 1/6-comma meantone is a 67-note set.
The 8ve-invariant 67th generator is ~9.168509182 lower (narrower)
than the starting pitch, and its tuning is 3^(67/3) * 5^(67/6).
The ratio it implies acoustically most closely is 3^23 * 5^11.
The unison-vector would therefore be described, in my matrix
notation, as (-61 23 11).

Gene, does this agree with your program's output?

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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

> The next "closure" size for 1/6-comma meantone is a 67-note set.
> The 8ve-invariant 67th generator is ~9.168509182 lower (narrower)
> than the starting pitch, and its tuning is 3^(67/3) * 5^(67/6).
> The ratio it implies acoustically most closely is 3^23 * 5^11.
> The unison-vector would therefore be described, in my matrix
> notation, as (-61 23 11).
>
> Gene, does this agree with your program's output?

I'm not sure what your question means; however I can make the following comments:

(1) Presumably you meant the comma 2^62 3^(-23) 5^(-11)

(2) This is a 67-et comma; however, and much more significantly, it is a 65-et comma. It really doesn't work very well for anything *but* 65-et, in fact.

(3) For the associated linear temperament, we have a map

[ 0 1]
[-11 7]
[ 23 9]

The generator is 31.997/65, so this can be more or less equated with
32/65.

--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: monz <joemonz@y...>
> > To: <tuning-math@y...>
> > Sent: Tuesday, December 18, 2001 6:48 PM
> > Subject: [tuning-math] 55-tET (was: Re: inverse of matrix --> for
what?)
> >
> >
> > What I had in mind was that there should be a pair of
> > unison-vectors which defines the set of acoustically implied
> > ratios which I put on my lattice at
> >
<http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm>?
> > ... assuming, of course, that in the places where two
> > ratios are implied equally well/badly, only one can be chosen.
> >
> > I find that if I continue my diagram, the unison-vector that
> > "works" together with the syntonic comma (-4 4 -1 ) to close
> > the system at 55 tones, is the (-51 19 9 ). The 8ve-invariant
> > tuning of the 55th quasi-meantone pitch would be 3^19 * 5^(55/6),
> > which is ~10.38405963 cents higher than the starting pitch, and
> > the ratio it implies most closely is 3^19 * 5^9.
>
>
> Oops... my bad. Two errors here.
>
> That should say "The 8ve-invariant tuning of the 55th
> quasi-meantone *generator*...", calling the starting pitch
> the zero-th generator. And the tuning itself is wrong:
> it should be 3^(55/3) * 5^(55/6).

What you completely fail to mention is that you're actually using 1/6-
comma meantone. This was nowhere implied in your original question to
us.

But you can indeed define 55-tET with the two unison vectors (-4 4 -
1) and (-51 19 9), and in fact it's quite logical to do so, since
these intervals are only 21.51 cents and 13.97 cents in JI.

In fact, I've been wondering if there's an analogue to Minkowski
reduction which, instead of finding the simplest commatic unison
vectors, finds the _smallest_ ones that work, without torsion
(or "potential torsion"). This could be valuable to JI-oriented
theorists like Monz and Kraig. For example, it seems that for
Blackjack in the 7-limit, the answer might be 2401:2400 and
16875:16807 . . . Gene, does this make any sense?

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, December 19, 2001 11:11 AM
> Subject: [tuning-math] Re: 55-tET
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > Oops... my bad. Two errors here.
> >
> > That should say "The 8ve-invariant tuning of the 55th
> > quasi-meantone *generator*...", calling the starting pitch
> > the zero-th generator. And the tuning itself is wrong:
> > it should be 3^(55/3) * 5^(55/6).
>
> What you completely fail to mention is that you're actually using 1/6-
> comma meantone. This was nowhere implied in your original question to
> us.

Wow -- my bad again. Thanks for catching that, Paul.
I made reference to my webpage, so it should have been pretty
clear that I was referring to 1/6-comma meantone specifically,
as well as its analogue in 55-EDO... but you're right that I
should have made that clear in my post.

> But you can indeed define 55-tET with the two unison vectors
> (-4 4 -1) and (-51 19 9), and in fact it's quite logical to do so,
> since these intervals are only 21.51 cents and 13.97 cents in JI.

Paul, I've been surprised all along that you disagree so strongly
with my interpretations of meantone systems. Is this perhaps
bringing our individual conceptions a bit closer together?

> In fact, I've been wondering if there's an analogue to Minkowski
> reduction which, instead of finding the simplest commatic unison
> vectors, finds the _smallest_ ones that work, without torsion
> (or "potential torsion"). This could be valuable to JI-oriented
> theorists like Monz and Kraig. For example, it seems that for
> Blackjack in the 7-limit, the answer might be 2401:2400 and
> 16875:16807 . . . Gene, does this make any sense?

Yes, this does sound interesting to me... but I really don't know
what "Minkowski reduction" is...

-monz

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

> In fact, I've been wondering if there's an analogue to Minkowski
> reduction which, instead of finding the simplest commatic unison
> vectors, finds the _smallest_ ones that work, without torsion
> (or "potential torsion"). This could be valuable to JI-oriented
> theorists like Monz and Kraig. For example, it seems that for
> Blackjack in the 7-limit, the answer might be 2401:2400 and
> 16875:16807 . . . Gene, does this make any sense?

It makes sense, but I don't think it defines a unique interval.

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

> > But you can indeed define 55-tET with the two unison vectors
> > (-4 4 -1) and (-51 19 9), and in fact it's quite logical to do so,
> > since these intervals are only 21.51 cents and 13.97 cents in JI.
>
>
> Paul, I've been surprised all along that you disagree so strongly
> with my interpretations of meantone systems. Is this perhaps
> bringing our individual conceptions a bit closer together?

Well, I still hold strongly to the views that I expressed, but that
doesn't mean that there isn't some mathematics that could be useful
to you for fleshing out _your_ views, nor that I would be averse to
helping you with such mathematics.

> > In fact, I've been wondering if there's an analogue to Minkowski
> > reduction which, instead of finding the simplest commatic unison
> > vectors, finds the _smallest_ ones that work, without torsion
> > (or "potential torsion"). This could be valuable to JI-oriented
> > theorists like Monz and Kraig. For example, it seems that for
> > Blackjack in the 7-limit, the answer might be 2401:2400 and
> > 16875:16807 . . . Gene, does this make any sense?
>
>
> Yes, this does sound interesting to me... but I really don't know
> what "Minkowski reduction" is...

That's means finding the _simplest_ (smallest numbers in the ratio)
commatic unison vectors defining a system. For 55-tET, the simplest
pair is . . . well, we gave it to you yesterday. One member of that
pair was the syntonic comma, but the other member of that pair was 93
cents or something, which would make little sense in a JI view of the
55-tone system. Why would someone temper out a 93-cent interval if
much smaller intervals abound at closer distances in the lattice?
Well, my answer to that is that you're not using JI at all, you've
already tempered out the 81:80, so that the "93-cent interval" in
question is contracted to something far less than 93 cents. However,
in your view, you're continuing to reference a JI pitch set with the
1/1 in common with the tempered pitch set you're actually using (very
far-fetched, I think, but that is your view). So in your view, the 55
tones would be much better understood as the Fokker periodicity block
defined by the two unison vectors (-4 4 -1) and (-51 19 9). Since I'm
sure you're interested, here are the coordinates of these 55 tones in
the (3,5) lattice:

3 5
--- ----

-11 -4
-10 -4
-9 -4
-8 -4
-7 -4
-6 -4
-9 -3
-8 -3
-7 -3
-6 -3
-5 -3
-4 -3
-7 -2
-6 -2
-5 -2
-4 -2
-3 -2
-2 -2
-5 -1
-4 -1
-3 -1
-2 -1
-1 -1
0 -1
-3 0
-2 0
-1 0
0 0
1 0
2 0
3 0
0 1
1 1
2 1
3 1
4 1
5 1
2 2
3 2
4 2
5 2
6 2
7 2
4 3
5 3
6 3
7 3
8 3
9 3
6 4
7 4
8 4
9 4
10 4
11 4

And here are the pitch-heights, in cents, of these same 55 just tones:

0
19.553
39.105
72.626
92.179
111.73
131.28
164.8
184.36
203.91
223.46
243.02
276.54
294.13
296.09
313.69
333.24
366.76
386.31
405.87
425.42
458.94
478.49
498.04
517.6
537.15
570.67
590.22
609.78
629.33
662.85
682.4
701.96
721.51
741.06
774.58
794.13
813.69
833.24
866.76
886.31
903.91
905.87
923.46
956.98
976.54
996.09
1015.6
1035.2
1068.7
1088.3
1107.8
1127.4
1160.9
1180.4

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > In fact, I've been wondering if there's an analogue to Minkowski
> > reduction which, instead of finding the simplest commatic unison
> > vectors, finds the _smallest_ ones that work, without torsion
> > (or "potential torsion"). This could be valuable to JI-oriented
> > theorists like Monz and Kraig. For example, it seems that for
> > Blackjack in the 7-limit, the answer might be 2401:2400 and
> > 16875:16807 . . . Gene, does this make any sense?
>
> It makes sense, but I don't think it defines a unique interval.

Meaning you can always find smaller and smaller examples? Even if you
disallow "potential torsion"? What if you fix all the commas except
one, and just have to find the smallest candidate for the remaining
comma. Isn't that choice unique?

--- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > > But you can indeed define 55-tET with the two unison vectors
> > > (-4 4 -1) and (-51 19 9), and in fact it's quite logical to do
so,
> > > since these intervals are only 21.51 cents and 13.97 cents in
JI.

There's an even smaller unison vector you can use, which comes from
subtracting these two from one another:

(47 15 10) = 7.54 cents.

Now, combining this with the syntonic comma, we get the following
Fokker periodicity block, which should be even closer to 55-tET:

3 5
--- ---

-7 -5
-6 -5
-5 -5
-8 -4
-7 -4
-6 -4
-5 -4
-4 -4
-7 -3
-6 -3
-5 -3
-4 -3
-3 -3
-2 -3
-5 -2
-4 -2
-3 -2
-2 -2
-1 -2
-4 -1
-3 -1
-2 -1
-1 -1
0 -1
1 -1
-2 0
-1 0
0 0
1 0
2 0
-1 1
0 1
1 1
2 1
3 1
4 1
1 2
2 2
3 2
4 2
5 2
2 3
3 3
4 3
5 3
6 3
7 3
4 4
5 4
6 4
7 4
8 4
5 5
6 5
7 5

cents:

0
19.553
39.105
72.626
92.179
111.73
131.28
143.3
162.85
203.91
223.46
243.02
255.03
274.58
315.64
335.19
354.75
366.76
386.31
405.87
446.93
458.94
478.49
498.04
517.6
558.66
570.67
590.22
609.78
629.33
641.34
682.4
701.96
721.51
741.06
753.07
794.13
813.69
833.24
845.25
864.81
884.36
925.42
944.97
956.98
976.54
996.09
1037.1
1056.7
1068.7
1088.3
1107.8
1127.4
1160.9
1180.4

Meanwhile, combining the two smallest so far, (-51 19 9) and (47 15
10), leads to this, closer still to 55-tET, but more unlikely from a
JI standpoint:

3 5
--- ---

-16 -9
-14 -8
-13 -7
-12 -7
-11 -6
-10 -6
-10 -5
-9 -5
-8 -5
-8 -4
-7 -4
-6 -4
-7 -3
-6 -3
-5 -3
-4 -3
-5 -2
-4 -2
-3 -2
-2 -2
-4 -1
-3 -1
-2 -1
-1 -1
0 -1
-2 0
-1 0
0 0
1 0
2 0
0 1
1 1
2 1
3 1
4 1
2 2
3 2
4 2
5 2
4 3
5 3
6 3
7 3
6 4
7 4
8 4
8 5
9 5
10 5
10 6
11 6
12 7
13 7
14 8
16 9

cents:

0
19.553
39.105
72.626
92.179
111.73
131.28
150.84
170.39
203.91
223.46
243.02
262.57
282.12
308.1
327.66
347.21
366.76
386.31
405.87
439.39
458.94
478.49
498.04
517.6
551.12
570.67
590.22
609.78
629.33
648.88
682.4
701.96
721.51
741.06
760.61
794.13
813.69
833.24
852.79
872.34
891.9
917.88
937.43
956.98
976.54
996.09
1029.6
1049.2
1068.7
1088.3
1107.8
1127.4
1160.9
1180.4

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

> > It makes sense, but I don't think it defines a unique interval.
>
> Meaning you can always find smaller and smaller examples? Even if you
> disallow "potential torsion"? What if you fix all the commas except
> one, and just have to find the smallest candidate for the remaining
> comma. Isn't that choice unique?

That should do it, though it seems a little arbitary. If you don't fix all but one, I can prove easily enough you get arbitarily small commas, so this may be your best shot. For the 55-et, or anything else where there is a clear set of all-but-one keepers in the comma department, it would make sense.

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "paulerlich" <paul@s...> wrote:
>
> > > It makes sense, but I don't think it defines a unique interval.
> >
> > Meaning you can always find smaller and smaller examples? Even if
you
> > disallow "potential torsion"? What if you fix all the commas
except
> > one, and just have to find the smallest candidate for the
remaining
> > comma. Isn't that choice unique?
>
> That should do it, though it seems a little arbitary. If you don't
>fix all but one, I can prove easily enough you get arbitarily small
>commas, so this may be your best shot. For the 55-et, or anything
>else where there is a clear set of all-but-one keepers in the comma
>department, it would make sense.

Well? Did I find it? 3^15*5^10.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, December 19, 2001 1:36 PM
> Subject: [tuning-math] Re: 55-tET
>
>
> Well, I still hold strongly to the views that I expressed, but that
> doesn't mean that there isn't some mathematics that could be useful
> to you for fleshing out _your_ views, nor that I would be averse to
> helping you with such mathematics.

Thanks! Much appreciated.

> ... So in your [monz's] view, the 55
> tones would be much better understood as the Fokker periodicity block
> defined by the two unison vectors (-4 4 -1) and (-51 19 9). Since I'm
> sure you're interested, here are the coordinates of these 55 tones in
> the (3,5) lattice:
>
> <table snipped>

Thanks, Paul! I haven't checked yet, but my guess is that the ratios
you provided here should be the same as the lattice that could be
extended from the one on my webpage (except for the commatic duplicated
tones on my lattice), yes?
http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, December 19, 2001 2:02 PM
> Subject: [tuning-math] Re: 55-tET
>
>
> There's an even smaller unison vector you can use, which comes from
> subtracting these two from one another:
>
> (47 15 10) = 7.54 cents.
>
> Now, combining this with the syntonic comma, we get the following
> Fokker periodicity block, which should be even closer to 55-tET:
>
> <table snipped>
>
> Meanwhile, combining the two smallest so far, (-51 19 9) and (47 15
> 10), leads to this, closer still to 55-tET, but more unlikely from a
> JI standpoint:
>
> <table snipped>

OK, now I have checked, and yes indeed, all the tables you've
provided are related to the lattice on my webpage.
http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm

In fact, this is quite interesting... as your tables describe
successively closer approximations to 55-tET, they also successively
eliminate the "commatic-duplicate" pitches on my lattice! In other
words, the implied ratios of the lattice *as a group* huddle closer
and closer to the linear axis which represents the actual meantone.
Hmmm....

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Wednesday, December 19, 2001 2:02 PM
> > Subject: [tuning-math] Re: 55-tET
> >
> >
> > There's an even smaller unison vector you can use, which comes
from
> > subtracting these two from one another:
> >
> > (47 15 10) = 7.54 cents.
> >
> > Now, combining this with the syntonic comma, we get the following
> > Fokker periodicity block, which should be even closer to 55-tET:
> >
> > <table snipped>
> >
> > Meanwhile, combining the two smallest so far, (-51 19 9) and (47
15
> > 10), leads to this, closer still to 55-tET, but more unlikely
from a
> > JI standpoint:
> >
> > <table snipped>
>
>
> OK, now I have checked, and yes indeed, all the tables you've
> provided are related to the lattice on my webpage.
> http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm

Related to? Look carefully -- the third table has some pretty "out-
there" pitches in the lattice.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Thursday, December 20, 2001 11:26 AM
> Subject: [tuning-math] Re: 55-tET
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > > From: paulerlich <paul@s...>
> > > To: <tuning-math@y...>
> > > Sent: Wednesday, December 19, 2001 2:02 PM
> > > Subject: [tuning-math] Re: 55-tET
> > >
> > >
> > > There's an even smaller unison vector you can use, which
> > > comes from subtracting these two from one another:
> > >
> > > (47 15 10) = 7.54 cents.
> > >
> > > Now, combining this with the syntonic comma, we get the following
> > > Fokker periodicity block, which should be even closer to 55-tET:
> > >
> > > <table snipped>
> > >
> > > Meanwhile, combining the two smallest so far, (-51 19 9) and
> > > (47 15 10), leads to this, closer still to 55-tET, but more
> > > unlikely from a JI standpoint:
> > >
> > > <table snipped>
> >
> >
> > OK, now I have checked, and yes indeed, all the tables you've
> > provided are related to the lattice on my webpage.
> > http://www.ixpres.com/interval/monzo/meantone/lattices/lattices.htm
>
> Related to? Look carefully -- the third table has some pretty "out-
> there" pitches in the lattice.

You're right about that, Paul. But that's only because 1/6-comma
meantone is only an approximation to 55-EDO, and of course as you
are well aware there are better fraction-of-a-comma approximations
to 55-EDO. I give some near the top of my 55-EDO page:
http://www.ixpres.com/interval/monzo/55edo/55edo.htm

The ratios given in your third table actually are related to the
lattice I drew for 1/6-comma, the differences becoming more
significant the farther one goes along the meantone chain from 1/1.

However, your periodicity-block comes pretty close to the lattice
that I started drawing as being implied by 3/17-comma meantone,
and I can see that it would be even closer to that implied by
10/57-comma meantone.

So yes, the mathematics you're doing here to calculate
periodicity-blocks from unison-vectors *does* relate to my
"acoustically implied ratios" lattices for meantones.

(I'll upload the lattices of the other fraction-of-a-comma
approximations to 55-EDO when I'm finished with them.
For now, just trust me...)

love / peace / harmony ...

-monz
http://www.monz.org
"All roads lead to n^0"

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