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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