ET that does adaptive-JI?

In a discussion from August, in which Bob Wendell was
looking for a nice consistent ET to supplement the use
of cents and the consensus was the 111-tET fit his
criteria, Paul wrote:

> tuning-math message 837
> From: "Paul Erlich" <paul@s...>
> Date: Thu Aug 23, 2001 5:18 pm
> Subject: Re: EDO consistency and accuracy tables (was: A little
research...)
>
>
> Right . . . but I think the whole idea of 72 or 111 or 121 as a least-
> common-denominator way of describing ideal musical practice kind of
> falters when _adaptive JI_ comes into the picture . . . doesn't it?

I see that there are follow-ups which I haven't yet read,
but in case they don't quite address this question, I'm
very curious -- what kind of criteria would have to be set
to find an ET that *does* work for adaptive-JI?

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> In a discussion from August, in which Bob Wendell was
> looking for a nice consistent ET to supplement the use
> of cents and the consensus was the 111-tET fit his
> criteria, Paul wrote:
>
>
> > tuning-math message 837
> > From: "Paul Erlich" <paul@s...>
> > Date: Thu Aug 23, 2001 5:18 pm
> > Subject: Re: EDO consistency and accuracy tables (was: A little
> research...)
> >
> >
> > Right . . . but I think the whole idea of 72 or 111 or 121 as a
least-
> > common-denominator way of describing ideal musical practice kind
of
> > falters when _adaptive JI_ comes into the picture . . . doesn't
it?
>
>
> I see that there are follow-ups which I haven't yet read,
> but in case they don't quite address this question, I'm
> very curious -- what kind of criteria would have to be set
> to find an ET that *does* work for adaptive-JI?
>
>
>
> -monz

Thanks for asking this now, Monz.

I have offered 152-tET as a Universal Tuning -- one reason for this
is that it supports the wonderful adaptive JI system of two (or
three, or more, if necessary) 1/3-comma meantone chains, tuned 1/3-
comma apart. This gives you 5-limit adaptive JI with no drift
problems, and the pitch shifts reduced to normally imperceptible
levels. 1/152 oct. ~= 1/150 oct. = 8 cents.

In addition, it acts as a strict 11-limit JI system, with the maximum
error in the consonant intervals and in the Tonality Diamond pitches
is 2.23 cents -- 0.68 cents through the 5-limit.

Finally, and most importantly, it contains 76-tET, which contains all
the linear temperaments that interest me most, as they support
omnitetrachordal scales (and are 7-limit):
1. Meantone within 19-tET subsets
2. Pajara
3. Double-Diatonic within 38-tET subsets
as well as the non-tetrachordal
4. Kleismic within 19-tET subsets (see
http://www.uq.net.au/~zzdkeena/Music/ChainOfMinor3rds.htm).
The full 152-tET could probably support a large number of other
interesting linear temperaments.

Anyway, this is all just an example of how ETs can be used
consistently by exploiting their inconsistency ;) though 152-tET _is_
consistent through the 11-limit (see above).

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, January 30, 2002 8:08 PM
> Subject: [tuning-math] Re: ET that does adaptive-JI?
>
>
> I have offered 152-tET as a Universal Tuning -- one reason for this
> is that it supports the wonderful adaptive JI system of two (or
> three, or more, if necessary) 1/3-comma meantone chains, tuned 1/3-
> comma apart. This gives you 5-limit adaptive JI with no drift
> problems, and the pitch shifts reduced to normally imperceptible
> levels. 1/152 oct. ~= 1/150 oct. = 8 cents.

Would those be equidistant chains of 1/3-comma MT? Or is
there some special interval between chains? Does it also
work equivalently as chains of 19-EDO, since that's so
close to 1/3-comma MT? I find this really interesting.

-monz

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> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, January 30, 2002 8:08 PM
> Subject: [tuning-math] Re: ET that does adaptive-JI?
>
>
> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
>
> > I see that there are follow-ups which I haven't yet read,
> > but in case they don't quite address this question, I'm
> > very curious -- what kind of criteria would have to be set
> > to find an ET that *does* work for adaptive-JI?
>
>
> Thanks for asking this now, Monz.
>
> I have offered 152-tET as a Universal Tuning -- one reason for this
> is that it supports the wonderful adaptive JI system of two (or
> three, or more, if necessary) 1/3-comma meantone chains, tuned 1/3-
> comma apart. This gives you 5-limit adaptive JI with no drift
> problems, and the pitch shifts reduced to normally imperceptible
> levels. 1/152 oct. ~= 1/150 oct. = 8 cents.
>
> In addition, it acts as a strict 11-limit JI system, with the maximum
> error in the consonant intervals and in the Tonality Diamond pitches
> is 2.23 cents -- 0.68 cents through the 5-limit.
>
> Finally, and most importantly, it contains 76-tET, which contains all
> the linear temperaments that interest me most, as they support
> omnitetrachordal scales (and are 7-limit):
> 1. Meantone within 19-tET subsets
> 2. Pajara
> 3. Double-Diatonic within 38-tET subsets
> as well as the non-tetrachordal
> 4. Kleismic within 19-tET subsets (see
> http://www.uq.net.au/~zzdkeena/Music/ChainOfMinor3rds.htm).
> The full 152-tET could probably support a large number of other
> interesting linear temperaments.
>
> Anyway, this is all just an example of how ETs can be used
> consistently by exploiting their inconsistency ;) though 152-tET _is_
> consistent through the 11-limit (see above).

Hmmm ... can you elaborate more on that bit about "exploiting
their inconsistency"? I don't quite get that.

Anyway, 152-EDO seems like a fantastic tuning to play around with!
Why have I read nothing about it before?

-monz

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--- In tuning-math@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning-math@y...>
> > Sent: Wednesday, January 30, 2002 8:08 PM
> > Subject: [tuning-math] Re: ET that does adaptive-JI?
> >
> >
> > I have offered 152-tET as a Universal Tuning -- one reason for
this
> > is that it supports the wonderful adaptive JI system of two (or
> > three, or more, if necessary) 1/3-comma meantone chains, tuned
1/3-
> > comma apart. This gives you 5-limit adaptive JI with no drift
> > problems, and the pitch shifts reduced to normally imperceptible
> > levels. 1/152 oct. ~= 1/150 oct. = 8 cents.
>
>
> Would those be equidistant chains of 1/3-comma MT?

Yes, equidistant at intervals of about 1/3-comma -- 1/152 oct. ~=
1/150 oct. = 8 cents.

> Or is
> there some special interval between chains?

Yes, about 1/3-comma -- 1/152 oct. ~= 1/150 oct. = 8 cents.

> Does it also
> work equivalently as chains of 19-EDO, since that's so
> close to 1/3-comma MT?

Right. Just like Vicentino's second tuning, where two 31-tET chains
1/4-comma apart could do all its tricks really well, here two (or
three, if you need certain chords besides major and minor triads) 19-
tET chains 1/3-comma apart do those same kinds of tricks.

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

> Hmmm ... can you elaborate more on that bit about "exploiting
> their inconsistency"? I don't quite get that.

Well, in most of the systems I described, you're using the second-
best or even third-best approximations to some consonant intervals,
relative to what 152-tET affords.

> Anyway, 152-EDO seems like a fantastic tuning to play around with!
> Why have I read nothing about it before?

Probably because such large tunings don't get mentioned much.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning-math@yahoogroups.com>
> Sent: Wednesday, January 30, 2002 9:13 PM
> Subject: [tuning-math] Re: ET that does adaptive-JI?
>
>

> --- In tuning-math@y..., "monz" <joemonz@y...> wrote:
> >
> > > From: paulerlich <paul@s...>
> > > To: <tuning-math@y...>
> > > Sent: Wednesday, January 30, 2002 8:08 PM
> > > Subject: [tuning-math] Re: ET that does adaptive-JI?
> > >
> > >
> > > I have offered 152-tET as a Universal Tuning -- one
> > > reason for this is that it supports the wonderful
> > > adaptive JI system of two (or three, or more, if necessary)
> > > 1/3-comma meantone chains, tuned 1/3-comma apart. This
> > > gives you 5-limit adaptive JI with no drift problems,
> > > and the pitch shifts reduced to normally imperceptible
> > > levels. 1/152 oct. ~= 1/150 oct. = 8 cents.
> >
> >
> > Would those be equidistant chains of 1/3-comma MT?
>
> Yes, equidistant at intervals of about 1/3-comma --
> 1/152 oct. ~= 1/150 oct. = 8 cents.
>
> > Or is there some special interval between chains?
>
> Yes, about 1/3-comma -- 1/152 oct. ~= 1/150 oct. = 8 cents.

Oh, sorry Paul ... I only noticed just now that you already said
"tuned 1/3-comma apart" in your original description. My bad.

> > Does it also work equivalently as chains of 19-EDO,
> > since that's so close to 1/3-comma MT?
>
> Right. Just like Vicentino's second tuning, where two 31-tET chains
> 1/4-comma apart could do all its tricks really well, here two (or
> three, if you need certain chords besides major and minor triads) 19-
> tET chains 1/3-comma apart do those same kinds of tricks.

Hmmm... and I see that the number 8 pops up again, because
152 = 19 * 8.

So 152-EDO is like 8 bicycle chains of 19-EDO, just like
72-EDO is like 6 bicycle chains of 12-EDO.

-monz

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