Yahoo Tuning Groups Ultimate Backup tuning-math 55-tET & 1/6-comma meantone

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Tuesday, December 18, 2001 8:25 PM
> Subject: [tuning-math] Re: 55-tET
>
>
> --- 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 [cents] 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)

Yes... I simply got the exponents from my lattice, and
inadvertently referenced them in the wrong direction.

> (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.

OK, here's what I really meant:

The 1/6-comma meantone generator = (3/2) / ( (81/80)^(1/6) ).

This is approximately equal to the following ET generators,
listed in order of increasing proximity of the ET generators
to the meantone one:

2^(~32.00865338 / 55)
2^(~38.99235958 / 67)
2^(~71.00101296 / 122)
2^(~322.9964114 / 555)
2^(~393.9974244 / 677)
2^(~464.9984373 / 799)

My idea was simply this: since 67-EDO approximates 1/6-comma
meantone better than 55-EDO, there should be a unison-vector
derived from 67-EDO which (along with 81:80) better defines
a periodicity-block for my "acoustically implied ratios" lattice
for 1/6-comma, than the one I got from 55-EDO, which was
(2^-51 * 3^19 * 5^9).

I would consequently suppose that the 2^(71/122) generator
results in a periodicity-block which is even closer to my
1/6-comma implied ratios lattice, and that 2^(323/555) is
closer still, etc. Yes?

I'm having a hard time following Gene's comments because
I don't understand why (2^62 * 3^-23 * 5^-11) "really doesn't
work very well for anything *but* 65-et" when in fact it
*is* also a 67-EDO comma.

...? Totally perplexed.

love / peace / harmony ...

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

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🔗genewardsmith <genewardsmith@juno.com>

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

> My idea was simply this: since 67-EDO approximates 1/6-comma
> meantone better than 55-EDO, there should be a unison-vector
> derived from 67-EDO which (along with 81:80) better defines
> a periodicity-block for my "acoustically implied ratios" lattice
> for 1/6-comma, than the one I got from 55-EDO, which was
> (2^-51 * 3^19 * 5^9).

If I LLL reduce the above pair I get 2^34 * 3^5 * 5^-18 for the second comma; TM reducing this then gives 2^38 * 3 * 5^-17. These are certainly better in the sense of simpler, though as commas the badness of the resulting temperaments is worse, since they are also quite a bit larger.

> I'm having a hard time following Gene's comments because
> I don't understand why (2^62 * 3^-23 * 5^-11) "really doesn't
> work very well for anything *but* 65-et" when in fact it
> *is* also a 67-EDO comma.

I thought when you asked me to run it through my program you meant to analyze the linear temperament it defines; from that point of view, the 67-et is too far out of tune to use it to much advantage, whereas the 65-et represents it well. Of course this temperament has nothing really to do with meantone.

🔗paulerlich <paul@stretch-music.com>

--- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > My idea was simply this: since 67-EDO approximates 1/6-comma
> > meantone better than 55-EDO, there should be a unison-vector
> > derived from 67-EDO which (along with 81:80) better defines
> > a periodicity-block for my "acoustically implied ratios" lattice
> > for 1/6-comma, than the one I got from 55-EDO, which was
> > (2^-51 * 3^19 * 5^9).
>
> If I LLL reduce the above pair I get 2^34 * 3^5 * 5^-18 for the
>second comma; TM reducing this then gives 2^38 * 3 * 5^-17. These
>are certainly better in the sense of simpler, though as commas the
>badness of the resulting temperaments is worse, since they are also
>quite a bit larger.

Gene, is this one of those mysterious magical facts? Or are you
forgetting that we are defining ETs here and not linear temperaments?

> > I'm having a hard time following Gene's comments because
> > I don't understand why (2^62 * 3^-23 * 5^-11) "really doesn't
> > work very well for anything *but* 65-et" when in fact it
> > *is* also a 67-EDO comma.
>
> I thought when you asked me to run it through my program you meant
> to analyze the linear temperament it defines;

That's not what Monz meant. Plus Monz has an additional conceptual
idiosyncracy -- he sees the JI scale implied _in toto_, not just via
consonant intervals, and thus the smaller (in cents) unison vectors
have a special meaning to him which to us, they do not.

🔗monz <joemonz@yahoo.com>

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, December 23, 2001 1:13 PM
> Subject: [tuning-math] Re: 55-tET & 1/6-comma meantone
>
>
> --- In tuning-math@y..., "genewardsmith" <genewardsmith@j...> wrote:
> > --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > > I'm having a hard time following Gene's comments because
> > > I don't understand why (2^62 * 3^-23 * 5^-11) "really doesn't
> > > work very well for anything *but* 65-et" when in fact it
> > > *is* also a 67-EDO comma.
> >
> > I thought when you asked me to run it through my program you meant
> > to analyze the linear temperament it defines;
>
> That's not what Monz meant. Plus Monz has an additional conceptual
> idiosyncracy -- he sees the JI scale implied _in toto_, not just via
> consonant intervals, and thus the smaller (in cents) unison vectors
> have a special meaning to him which to us, they do not.

Paul, I'm having a hard time understanding the difference between
these two conceptions, but I think I'm beginning to get it.

The implied ratios on my lattices follow the general trend of
the meantone axis itself, which implies a handful of intervals
which can be stacked to build the entire scale.

But some of these JI intervals are emphatically *not consonant*,
and are the "wolf intervals" which cause the displacement of
the trend-line of the periodicity-block to align it with the
meantone axis.

Am I on the right track?

-monz

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🔗paulerlich <paul@stretch-music.com>

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

> Paul, I'm having a hard time understanding the difference between
> these two conceptions, but I think I'm beginning to get it.
>
> The implied ratios on my lattices follow the general trend of
> the meantone axis itself, which implies a handful of intervals
> which can be stacked to build the entire scale.
>
> But some of these JI intervals are emphatically *not consonant*,
> and are the "wolf intervals" which cause the displacement of
> the trend-line of the periodicity-block to align it with the
> meantone axis.
>
> Am I on the right track?

No, not really. This doesn't have quite that much to do with the wolf
intervals, though it's related to the fact that once 81/80 is
tempered out, Gene (if I may speak for him) and I would view the
operative lattice as a cylindrical one -- the planar 2-d JI lattice
no longer applies to the tuning, musically, psychologically, or
spiritually.

🔗monz <joemonz@yahoo.com>

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Sunday, December 23, 2001 1:28 PM
> Subject: [tuning-math] Re: 55-tET & 1/6-comma meantone
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > Am I on the right track?
>
> No, not really. This doesn't have quite that much to do with the wolf
> intervals, though it's related to the fact that once 81/80 is
> tempered out, Gene (if I may speak for him) and I would view the
> operative lattice as a cylindrical one -- the planar 2-d JI lattice
> no longer applies to the tuning, musically, psychologically, or
> spiritually.

And I've agreed with both of you many times in the past, and wish
to emphasize again that the only reason I'm using a planar lattice
is because it's beyond my abilities to draw cylindrical ones. In
fact, I'd very much appreciate someone posting the mathematics for
converting my Excel lattices into cylindrical ones.

What would emerge from my meantone/JI-implication lattices if
they were to be "cylindrified", is that each flavor of meantone
would slice the cylinder diagonally at a different angle, and
would also impart a unique diameter to each cylinder. Right?

C'mon, guys... I'm itching to draw this stuff now...

-monz

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🔗paulerlich <paul@stretch-music.com>

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

> And I've agreed with both of you many times in the past, and wish
> to emphasize again that the only reason I'm using a planar lattice
> is because it's beyond my abilities to draw cylindrical ones.

It shouldn't be -- you can simply "ink" the cylinder and then "roll"
it a bunch of times over a flat sheet.

> In
> fact, I'd very much appreciate someone posting the mathematics for
> converting my Excel lattices into cylindrical ones.

Your Excel lattices, though, are currently referring to JI ratios,
including some rather complex ones -- we have to get rid of this
feature first. Gene, any clever ideas?
>
> What would emerge from my meantone/JI-implication lattices if
> they were to be "cylindrified", is that each flavor of meantone
> would slice the cylinder diagonally at a different angle, and
> would also impart a unique diameter to each cylinder. Right?

Hmm . . . not really. It seems to me that the angle and the diameter
would be fixed, and the _second_ unison vector tells you how _long_
the cylinder is before it meets itself, when bent into a torus
representing the ET you're approximating. If there is no second
unison vector, than the different flavors of meantone are
functionally identical, their only salient difference being the level
of beating in the consonant intervals.

It *would* be nice, I admit, to see some actual cylindrical
arrangements of notes, particularly in a VRML implementation or
something. So far, I've simply printed out flat, repeating lattices
and then rolled them up by hand.