Re: JI and the listening composer-reply to Carl

>And to clarify what Gene was suggesting, 612-equal gives 5-limit intervals
>to within a 20th of a cent and 9-limit to about a fifth of a cent. So
>that's very different to "within a cent or two". And you only need 10
>notes.

Thanks, Graham. And I'll note that the only instruments that I know of
that you can even tune with that degree of precision are electronic, or
very expensive pianos. So anyone claiming they can hear the difference
should say what they were doing when they heard a difference. Then I'd
be genuinely interested, as I was when Graham said he'd tuned
electronically-generated sawtooths to within 0.1 cents of a 3:2.

-Carl

In my sometimes humble opinion, errors of one of two cents are only
significant in aural bearing plans where tuning errors can accumulate
and cause larger discrepancies elsewhere. With electronic tuning
technology, even acoustic instruments can be tuned with this accuracy
and cumulative errors are not an issue with electronic tuning.

I challenge anyone to detect a one cent error even with unisons when
the pitches are not simultaneous and there is a gap of over half a
second between two consecutive pitches one cent apart. I can barely
hear two cents under these conditions and wouldn't notice it if I
weren't looking for it. If there are those who can meet this
challenge successfully, they are so rare as to be insignificant in
any practical musical scenario.

No one would be happier than I if such people abounded. I find my
ears constantly assailed by tin-eared "musicians" who miss with
pitches orders of magitude worse. They have, nevertheless, somehow
been contracted to perform for classical recordings on well-known
commercial labels.

So I would like to suggest that we have much bigger fires to fight
than squabbling over errors under two cents. One of the main
objections I hear to anything other than 12t-ET is that the
differences between 12t-ET and JI are not practically significant.

This burns me, but squabbling over errors under two cents when they
are not cumulative completely undermines any case we could make in
answer to such stupidity and classifies us to the common practice
world as anally afflicted theoretical nitpickers. Moreover, during
the infamous and strange Jerries experiments, I failed to see any
evidence among those of us who participated that anything like this
kind of aural acumen exists even among us microtonal enthusiasts.

And by the way, 612-EDO is incredibly accurate through 19-limit if we
don't remain "anal" about very small errors that are still larger
than "a fifth of a cent".

Cheers,

Bob

--- In tuning@y..., Carl Lumma <carl@l...> wrote:
> >And to clarify what Gene was suggesting, 612-equal gives 5-limit
intervals
> >to within a 20th of a cent and 9-limit to about a fifth of a
cent. So
> >that's very different to "within a cent or two". And you only
need 10
> >notes.
>
> Thanks, Graham. And I'll note that the only instruments that I
know of
> that you can even tune with that degree of precision are
electronic, or
> very expensive pianos. So anyone claiming they can hear the
difference
> should say what they were doing when they heard a difference. Then
I'd
> be genuinely interested, as I was when Graham said he'd tuned
> electronically-generated sawtooths to within 0.1 cents of a 3:2.
>
> -Carl

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:

> So I would like to suggest that we have much bigger fires to fight
> than squabbling over errors under two cents.

Errors under two cents are continually at issue on tuning-math, because an important consideration is what cutoff we should give in the search for temperaments. If people want microtemperaments, they are out there.

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
>
Bob originally:
> > So I would like to suggest that we have much bigger fires to
fight
> > than squabbling over errors under two cents.
>
Gene replied:
> Errors under two cents are continually at issue on tuning-math,
because an important consideration is what cutoff we should give in
the search for temperaments. If people want microtemperaments, they
are out there.

Bob now:
Well, I do like knowing they're out there, but with all due respect,
for me the cutoff point is the human capacity to hear the difference
and not some inaudible mathematical consideration. No one would love
it more than I if there were some relatively low number of equal
divisions per octave that would perfectly coincide with just
intervals through the nineteen limit.

However, given that the potential for finding this is infinitely less
than that for finding the Holy Grail, the engineering part of me says
we need to get practical and find a good palette of audibly "perfect"
and usable compromises. As long as the math serves this purpose, I'm
happy. Given that some of us are looking for practically accurate
finite models of JI, as soon as we lose sight of this purpose and get
lost in what seem to me to be musically insignificant mathematical
niceties, I feel we more than fulfill the worst expectations of our
severest critics among practicing musicians.

In the interest of making this discussion more practically accessible
to all interested parties, I present the following thoughts regarding
possible implementations:

72-tET for me is great through the 11-limit because it includes both
meantone capabilities and good JI approximations. It's good through
the 19-limit if you don't care about the primes 13 and 17 being 7.2
and 5.0 cents flat respectively, but I would never call errors of 5-7
cents inaudible. Also 19 is 2.5 cents sharp, but that wouldn't bother
me at all.

217 is interesting as a finite model for JI, but the lowest number of
steps per octave I know of for an ET that approximates everything
through 19-limit within one cent is 612. Here we are talking steps of
only 1.96 cents, so this approach is a bit brute force. The prime 17
is almost a half-step off in 612-EDO at 0.93 cents sharp.

However, if we choose our cutoff at 2.5 cents, then 152-EDO is great
for everything except prime 13 at 3.7 cents flat, also almost a half-
step in 152-EDO. For me it's a matter of trading this off with the
number of steps per octave required, and few people if any are so
sensitive to tunings involving the prime 13 that 3.7 cents flat is
going to bother their ears as opposed to their eyes perhaps on paper.

The worst error in 171-EDO through the 19-limit is 3.07 cents sharp
for the prime 11. Next is prime 19 at 2.8 cents flat. All the rest
are under 2 cents. So the tradeoff between 152- and 171-EDO is number
of steps per octave vs. a worst error difference of 0.62 cents.

So for me 152 is the choice of the moment for ET approximations of JI
through the 19-limit. I happened onto this after independently
discovering 217 as 7 X 31 playing with my HP scientific on the basis
of Vicento's experimentations. I then looked at 1/3-comma and 19-EDO
in the same way and saw that 8 X 19 = 152 and 9 X 19 = 171 both
looked quite good. I almost ignored this though, until Paul Erlich
said he was playing with 152-EDO. I looked into it further and really
liked it.

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:

> 217 is interesting as a finite model for JI, but the lowest number of
> steps per octave I know of for an ET that approximates everything
> through 19-limit within one cent is 612.

460 does this, and if you want an all-purpose microtemperament, the Old Reliable is the 311-et; it is off by 1.4 cents in the 19 limit, but is good to higher limits. Of course, the higher you go the less difference it makes what et you choose.

> However, if we choose our cutoff at 2.5 cents, then 152-EDO is great
> for everything except prime 13 at 3.7 cents flat, also almost a half-
> step in 152-EDO.

The 121-et gets everything (not just primes) in the 19-limit within 4.35 cents, and is worth considering.

For me it's a matter of trading this off with the
> number of steps per octave required, and few people if any are so
> sensitive to tunings involving the prime 13 that 3.7 cents flat is
> going to bother their ears as opposed to their eyes perhaps on paper.

There's a theory that higher prime limits require higher accuracy, which seems plausible, don't you think?

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
> --- In tuning@y..., "genewardsmith" <genewardsmith@j...>
wrote:
> > --- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
> >
> Bob originally:
> > > So I would like to suggest that we have much bigger fires to
> fight
> > > than squabbling over errors under two cents.
> >
> Gene replied:
> > Errors under two cents are continually at issue on
tuning-math,
> because an important consideration is what cutoff we should
give in
> the search for temperaments. If people want
microtemperaments, they
> are out there.
>
> Bob now:
> Well, I do like knowing they're out there, but with all due
respect,
> for me the cutoff point is the human capacity to hear the
difference
> and not some inaudible mathematical consideration. No one
would love
> it more than I if there were some relatively low number of equal
> divisions per octave that would perfectly coincide with just
> intervals through the nineteen limit.
>
> However, given that the potential for finding this is infinitely less
> than that for finding the Holy Grail, the engineering part of me
says
> we need to get practical and find a good palette of audibly
"perfect"
> and usable compromises. As long as the math serves this
purpose, I'm
> happy. Given that some of us are looking for practically
accurate
> finite models of JI, as soon as we lose sight of this purpose
and get
> lost in what seem to me to be musically insignificant
mathematical
> niceties, I feel we more than fulfill the worst expectations of our
> severest critics among practicing musicians.
>
> In the interest of making this discussion more practically
accessible
> to all interested parties, I present the following thoughts
regarding
> possible implementations:
>
> 72-tET for me is great through the 11-limit because it includes
both
> meantone capabilities and good JI approximations. It's good
through
> the 19-limit if you don't care about the primes 13 and 17 being
7.2
> and 5.0 cents flat respectively, but I would never call errors of
5-7
> cents inaudible. Also 19 is 2.5 cents sharp, but that wouldn't
bother
> me at all.
>
> 217 is interesting as a finite model for JI, but the lowest
number of
> steps per octave I know of for an ET that approximates
everything
> through 19-limit within one cent is 612. Here we are talking
steps of
> only 1.96 cents, so this approach is a bit brute force. The prime
17
> is almost a half-step off in 612-EDO at 0.93 cents sharp.
>
> However, if we choose our cutoff at 2.5 cents, then 152-EDO is
great
> for everything except prime 13 at 3.7 cents flat, also almost a
half-
> step in 152-EDO. For me it's a matter of trading this off with the
> number of steps per octave required, and few people if any are
so
> sensitive to tunings involving the prime 13 that 3.7 cents flat is
> going to bother their ears as opposed to their eyes perhaps on
paper.
>
> The worst error in 171-EDO through the 19-limit is 3.07 cents
sharp
> for the prime 11. Next is prime 19 at 2.8 cents flat. All the rest
> are under 2 cents. So the tradeoff between 152- and 171-EDO
is number
> of steps per octave vs. a worst error difference of 0.62 cents.
>
> So for me 152 is the choice of the moment for ET
approximations of JI
> through the 19-limit. I happened onto this after independently
> discovering 217 as 7 X 31 playing with my HP scientific on the
basis
> of Vicento's experimentations. I then looked at 1/3-comma and
19-EDO
> in the same way and saw that 8 X 19 = 152 and 9 X 19 = 171
both
> looked quite good. I almost ignored this though, until Paul
Erlich
> said he was playing with 152-EDO. I looked into it further and
really
> liked it.

Wow woud you really use a EDO like 152. That seems like way
too many notes to be practical.

What is the purpose in mind? Electronic composition, acoustic
ensembles, voices?

I think you would need a pretty serious keyboard like the
microzone to get much real time access to those notes.

Why not just go for straight Just Intonation and only use the
notes that are required for your composition. I do not see any
need for a all purpose holy grail tuning. We already had one of
those [whose shackles we have barely begun to remove].

Soon enough electronic instruments will probably introduce
adptive tuning models. The thing to speed tis along is to get fols
interested in listening to music that is otherly tuned.

Is this desire for an EDO perhaps a notation thing ?

Justin White

I personally am looking for simplified, finite conceptual tools or
models that approximate JI accurately enough to make little if any
difference to the human ear and yet make things accessible to
everyone who wishes to compose in something closely approximating JI.

Pure JI composition is problematic for a lot of musicians who may not
want to get into calculations as deeply as it requires. A simple
table of interval definitions in terms of ET steps would do in the
context of any good n-EDO model. I hope this is clear. Keyboards are
an entirely separate and unrelated issue in this context.

Implementation of the compositions may or may not involve keyboards,
but not necessarily ones with 72 or 121+ notes per octave.

Cheers,

Bob

--- In tuning@y..., "justintonation" <JUSTINTONATION@H...> wrote:
>
>
> Wow woud you really use a EDO like 152. That seems like way
> too many notes to be practical.
>
> What is the purpose in mind? Electronic composition, acoustic
> ensembles, voices?
>
> I think you would need a pretty serious keyboard like the
> microzone to get much real time access to those notes.
>
> Why not just go for straight Just Intonation and only use the
> notes that are required for your composition. I do not see any
> need for a all purpose holy grail tuning. We already had one of
> those [whose shackles we have barely begun to remove].
>
> Soon enough electronic instruments will probably introduce
> adptive tuning models. The thing to speed tis along is to get fols
> interested in listening to music that is otherly tuned.
>
> Is this desire for an EDO perhaps a notation thing ?
>
> Justin White

--- In tuning@y..., "justintonation" <JUSTINTONATION@H...> wrote:

> Why not just go for straight Just Intonation and only use the
> notes that are required for your composition.

that wouldn't work for me even for a simple diatonic scale. but i
only speak for myself!

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
> I personally am looking for simplified, finite conceptual tools or
> models that approximate JI accurately enough to make little if any
> difference to the human ear and yet make things accessible to
> everyone who wishes to compose in something closely approximating
> JI.

for this, you have to specify whether you mean strict ji or adaptive
ji, and which commas you expect not to vanish. we can pursue this
some more on tuning-math if you like.

Hello robert!
The calculations of JI intervals can be done by at 8 year old. Also you can hear them. The use
of square roots to x power are more complex calculations and seems more problematic.

robert_wendell wrote:

>
> Pure JI composition is problematic for a lot of musicians who may not
> want to get into calculations as deeply as it requires. A simple
> table of interval definitions in terms of ET steps would do in the
> context of any good n-EDO model. I

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

Yes, I'm aware of this, but the same problem exists whether you
employ pure JI or a finite ET model approximating it.

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
>
> for this, you have to specify whether you mean strict ji or
adaptive
> ji, and which commas you expect not to vanish. we can pursue this
> some more on tuning-math if you like.

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> Hello robert!
> The calculations of JI intervals can be done by at 8 year old.
Also you can hear them. The use
> of square roots to x power are more complex calculations and seems
more problematic.
>

Bob:
There are a lot of musicians who cannot do third grade arithmetic
accurately, much less easily grasp the concepts behind the JI
calculations. The nth roots of two (I think you meant) can be done on
any scientific calculator: presto, whamo! Also, even that is not
necessary if you have a predefined palette of pitches to work with
that you got from someone more technically inclined or some
commercial software vendor.

In a message dated 6/15/02 9:31:07 PM Eastern Daylight Time,
rwendell@cangelic.org writes:

> Bob:
> There are a lot of musicians who cannot do third grade arithmetic
> accurately, much less easily grasp the concepts behind the JI
> calculations.

I believe, then, that this is an education issue. Rather than sweep it under
the rug shouldn't singing just intervals be part of a basic education?

Johnny Reinhard

--- In tuning@y..., Afmmjr@a... wrote:
> In a message dated 6/15/02 9:31:07 PM Eastern Daylight Time,
> rwendell@c... writes:
>
>
> > Bob:
> > There are a lot of musicians who cannot do third grade arithmetic
> > accurately, much less easily grasp the concepts behind the JI
> > calculations.
>
> I believe, then, that this is an education issue. Rather than
sweep it under
> the rug shouldn't singing just intervals be part of a basic
education?
>
> Johnny Reinhard

Bob:
Of course! But I don't understand what singing just intervals has to
do with calculating JI intervals for purposes of composition.

If you know anything about me from my posts here in the past, you
know that I dedicate myself musically to teaching singers to sing
just intervals. I use pure JI to do that. Why mess with temperaments
when you're just dealing mostly with perfect fourths and fifths and
thirds during tuning exercises?

I've outlined my whole approach in detail in other posts. I have
found it highly valuable and efficient at any level of singing
ability. Go to Files and look in my folder.
You will find a sound file of my choir singing in JI pretty darn
accurately most of the time.

/tuning/files/Bob%20Wendell

It's not even the best example, but one small enough to upload to
this list. For more and better, go to

http://www.cangelic.org

and listen to the files from both CDs. This is with a choir in a town
of 10,000 in SE Iowa where the closest towns more than one tenth our
size are a half hour away. Lots of hogs, corn and beans,
though!...but not much intonation that even approaches 12t-ET for
accuracy except in my choir.

In a message dated 6/15/02 10:41:15 PM Eastern Daylight Time,
rwendell@cangelic.org writes:

> Of course! But I don't understand what singing just intervals has to
> do with calculating JI intervals for purposes of composition.
>
>

I'm sure you are getting great results. Personally, drift doesn't bother me
at all. Maybe it is because I do not have perfect pitch. To me it is just
another dimension. Early Renaissance music works beautifully in just.

As for compositional reasons, I'm not sure I understand you. If you mean to
have a temperament that allows for easy planning, I guess that works. But I
don't really see the issue because an avid Just composer will certainly work
out the math. Why a secondary translation of Just through any temperament at
all?

Kind regards, Johnny Reinhard

we're not quite on the same wavelength. in pure ji the commas are
what they are, and none of them vanish . . . so i'm not sure what
you're thinking of as the "problem" here . . . reply off-list if this
is a simple miscommunication . . .

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
> Yes, I'm aware of this, but the same problem exists whether you
> employ pure JI or a finite ET model approximating it.
>
> --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> >
> > for this, you have to specify whether you mean strict ji or
> adaptive
> > ji, and which commas you expect not to vanish. we can pursue this
> > some more on tuning-math if you like.

Hello Robert!
You can use the same calculator to do the easier math of JI. You could just as easily get a
set of JI pitches from someone else. a 7/6 above 440? try times 7 divided by 6.

robert_wendell wrote:

> --- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> > Hello robert!
> > The calculations of JI intervals can be done by at 8 year old.
> Also you can hear them. The use
> > of square roots to x power are more complex calculations and seems
> more problematic.
> >
>
> Bob:
> There are a lot of musicians who cannot do third grade arithmetic
> accurately, much less easily grasp the concepts behind the JI
> calculations. The nth roots of two (I think you meant) can be done on
> any scientific calculator: presto, whamo! Also, even that is not
> necessary if you have a predefined palette of pitches to work with
> that you got from someone more technically inclined or some
> commercial software vendor.
>

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., Afmmjr@a... wrote:
> In a message dated 6/15/02 10:41:15 PM Eastern Daylight Time,
> rwendell@c... writes:
>
>
> > Of course! But I don't understand what singing just intervals has
to
> > do with calculating JI intervals for purposes of composition.
> >
> >
>
> I'm sure you are getting great results. Personally, drift doesn't
bother me
> at all. Maybe it is because I do not have perfect pitch. To me it
is just
> another dimension. Early Renaissance music works beautifully in
just.
>
> As for compositional reasons, I'm not sure I understand you. If
you mean to
> have a temperament that allows for easy planning, I guess that
works. But I
> don't really see the issue because an avid Just composer will
certainly work
> out the math. Why a secondary translation of Just through any
temperament at
> all?
>
> Kind regards, Johnny Reinhard

Bob:
Well, there sure has been a lot of time wasted on this tuning list
discussing just that if it's indeed a useless pursuit. I personally
am pretty technically oriented, being a musician and ex-engineer.
Nontheless, I believe it would be a boon to the freedom of my
creative expression to have a relatively finite and simple near-JI
model with equal steps to play with in my planning, at least to
start. Even then, if I decided it was worth the trouble, I could
refine it by converting to a pure JI version.

Well, I'm not sure that it is a simple miscommunication. You had
stated that some comma problems still have to be dealt with in
temperaments that accurately approximate JI. I agreed, 53-EDO being a
good example. 72 provides some interesting options for compromise,
such as a short closed cycle identical to 12-EDO if you choose to use
it.

One could do cyclical modulations based on 12-EDO, but maintain
vertical structures that are quasi-just or any combination of that
with other techniques. I can imagine wanting to use the 12-EDO for
quartal harmonic textures, for example, then switching to or
alternating with triadic and seventh-chord textures using more quasi-
just structures. This kind of compositional ease of creative
expression is a major motive for me in looking at finite, quasi-just
ET models, to address an issue that has arisen in another thread.

Cheers,

Bob

--- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> we're not quite on the same wavelength. in pure ji the commas are
> what they are, and none of them vanish . . . so i'm not sure what
> you're thinking of as the "problem" here . . . reply off-list if
this
> is a simple miscommunication . . .
>
> --- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
> > Yes, I'm aware of this, but the same problem exists whether you
> > employ pure JI or a finite ET model approximating it.
> >
> > --- In tuning@y..., "emotionaljourney22" <paul@s...> wrote:
> > >
> > > for this, you have to specify whether you mean strict ji or
> > adaptive
> > > ji, and which commas you expect not to vanish. we can pursue
this
> > > some more on tuning-math if you like.

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:

> Well, I'm not sure that it is a simple miscommunication. You had
> stated that some comma problems still have to be dealt with in
> temperaments that accurately approximate JI.

that's in the context of common-practice, diatonic music.

> I agreed, 53-EDO being a
> good example. 72 provides some interesting options for compromise,
> such as a short closed cycle identical to 12-EDO if you choose to
use
> it.

as well as other short closed cycles, such as those implied by the
blackjack lattice.

> One could do cyclical modulations based on 12-EDO, but maintain
> vertical structures that are quasi-just or any combination of that
> with other techniques. I can imagine wanting to use the 12-EDO for
> quartal harmonic textures, for example, then switching to or
> alternating with triadic and seventh-chord textures using more
quasi-
> just structures. This kind of compositional ease of creative
> expression is a major motive for me in looking at finite, quasi-
just
> ET models, to address an issue that has arisen in another thread.
>
> Cheers,
>
> Bob

and music notated in this way would likely be readily understood by
performers. given the success you've had bringing more conventially
notated compositions to life in adaptive ji, i can only look forward
to the ultimate results of this with the highest of eagerness!

--- In tuning@y..., "robert_wendell" <rwendell@c...> wrote:
> I personally am looking for simplified, finite conceptual tools or
> models that approximate JI accurately enough to make little if any
> difference to the human ear and yet make things accessible to
> everyone who wishes to compose in something closely approximating
JI.
>
> Pure JI composition is problematic for a lot of musicians who may
not
> want to get into calculations as deeply as it requires. A simple
> table of interval definitions in terms of ET steps would do in the
> context of any good n-EDO model. I hope this is clear. Keyboards
are
> an entirely separate and unrelated issue in this context.
>
> Implementation of the compositions may or may not involve
keyboards,
> but not necessarily ones with 72 or 121+ notes per octave.
>
> Cheers,
>
> Bob
>
> --- In tuning@y..., "justintonation" <JUSTINTONATION@H...> wrote:
> >
> > Wow woud you really use a EDO like 152. That seems like way
> > too many notes to be practical.
> >
> > What is the purpose in mind? Electronic composition, acoustic
> > ensembles, voices?
> >
> > ... Is this desire for an EDO perhaps a notation thing ?
> >
> > Justin White

Bob,

It looks to me as if it is as at least as much of a notation thing as
anything else. While Gene offered some suggestions, nobody really
nailed this one down.

Our sagittal notation project over on tuning-math has as one of its
primary objectives notating just intonation to at least the 19
limit. We have a solution that allows this to be accomplished in two
different ways:

1) Provision of symbols for rational intervals, independent of any
division of the octave. Inasmuch as the number of symbols in the
notation is not infinite, it is conceivable that, using this
approach, one could run out of symbols in some cases.

2) Provision of symbols for a large-numbered ET that would allow
consistent and unique mapping of all 19-limit consonances. This
approach would guarantee that one would never run out of symbols, but
it is possible that two different rational intervals might be notated
the same way (although there are ways around this).

What it comes down to, then, is finding the ET of lowest number that:

1) Is 19-limit consistent;

2) Is 19-limit unique; and

3) Does not have an excessive error for any odd number in the 19
limit.

For the 11 limit, the solution would be 72-ET. For the 19 limit, the
solution is 217-ET. (193-ET comes close, but no cigar!)

There is a parallel between the two: 72 is 12 times 6; 72-ET is
achievable with considerable pitch bending (up to 50 cents) on some
12-ET instruments. 217 is 31 times 7; 217-ET would be achievable
with a relatively small amount of pitch bending (up to ~17 cents) on
31-ET instruments, should they be constructed.

The maximum deviation for 19-limit consonances in 217-ET is ~2.76
cents, with the typical deviation around half of that. Some
(including myself) will find this acceptable; some will not. (I
prefer not to argue the point; there is also the rational notation
approach for those who demand greater precision.)

The bottom line is that 217-ET does offer a framework such as you
desire for conceptualizing 19-limit intervals that could ultimately
find a way into practical usage via future 31-ET instruments.

--George

I'm not sure you ever saw this, Johnny, so I've edited and added to
it and am sending it directly to you as well as reposting it on the
tuning list.

Cheers,

Bob

Johnny R.:
> > As for compositional reasons, I'm not sure I understand you. If
> you mean to
> > have a temperament that allows for easy planning, I guess that
> works. But I
> > don't really see the issue because an avid Just composer will
> certainly work
> > out the math. Why a secondary translation of Just through any
> temperament at
> > all?
> >
> > Kind regards, Johnny Reinhard
>
> Bob:
> Well, there sure has been a lot of time wasted on this tuning list
> discussing ET approximations to just that if it's indeed a useless
pursuit. I personally am pretty technically oriented as a musician
and ex-engineer. Nonetheless, I believe it would be a boon to the
freedom of my creative expression to have a finite and relatively
simple quasi-JI model with equal steps to play with in my planning,
at least to start. Then, if I decided it was worth the trouble, I
could refine it by converting to a pure JI version.

After all, there is a kind of improvisational quality, I believe, to
the best composition. Most of us are used to improvising in some kind
of 12-note environment and therefore have a certain fluency in that
kind of musical medium. That certainly is the case for me. So in the
case of JI, I'm not so interested in composition that is purely
intellectually crafted work, hopefully with some feeling and
spontaneity in the background to drive it. I would rather have a
certain sense of fluidity in the compositional process that would
increase the spontaneous, directly, interactively and musically
inspired component.