updated definition: "just intonation"

Hello all,

I've finally added a lattice-diagram of an extensive
5-limit JI system to my Tuning Dictionary entry for
"just intonation":

http://www.ixpres.com/interval/dict/just.htm

This is something that should have been there a long
time ago.

I'm bringing it to the attention of Tuning List
subscribers so that anyone who wants to illustrate a
particular 5-limit rectangular lattice may do so by
simply modifying my .gif file.

That is ... if you like the angles I use in my lattice
formula, which are different from everyone else's.
My rational for choosing the angles I use is given here:
http://www.ixpres.com/interval/monzo/lattices/lattices.htm

(That page is also available in French and German.
Note also that I now use an inverted version of my
formula, so that angles are calculated from the 12 o'clock
position rather than 6 o'clock.)

-monz

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

When I came on these groups and used JI to mean rational intonation, I was told I was wrong. Your definition says that I was right, but also that by preference it means 5-limit, which I can't recall ever hearing. The situation seems thoughly snafued.

Joe!
I once again object to this definition in that the greek theory is universally recognized as using
Just intonation long before the idea of limits came along. To have to refer to them as Extended
just intonation is absurd. It also ignores that there is already the historical use by composers
about their own music such Partch and Harrison who refer to their music as Just Intonation
(including myself). To try to change the terms now is just too late and it does nothing to
further the dialog except to ghettoize the actual practitioners into being Extreme as in extended
and out side of the norm. Also outside of a theoretical novelty the actual cultures that use 5
limit system do not an have not referred to it as Just intonation.
I find the occasional impulse to change terminology already in existence on this list fails to
recognize the limits of this list. It would benefit from avoiding the situation of degenerating
into being a self proclaimed authority having say as to what occurs outside its boundaries

monz wrote:

> Hello all,
>
> I've finally added a lattice-diagram of an extensive
> 5-limit JI system to my Tuning Dictionary entry for
> "just intonation":
>
> http://www.ixpres.com/interval/dict/just.htm
>
> This is something that should have been there a long
> time ago.
>
> I'm bringing it to the attention of Tuning List
> subscribers so that anyone who wants to illustrate a
> particular 5-limit rectangular lattice may do so by
> simply modifying my .gif file.
>
> That is ... if you like the angles I use in my lattice
> formula, which are different from everyone else's.
> My rational for choosing the angles I use is given here:
> http://www.ixpres.com/interval/monzo/lattices/lattices.htm
>
> (That page is also available in French and German.
> Note also that I now use an inverted version of my
> formula, so that angles are calculated from the 12 o'clock
> position rather than 6 o'clock.)
>
> -monz
>
>
>
>

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

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

Hi Kraig,

> From: Kraig Grady <kraiggrady@anaphoria.com>
> To: <tuning@yahoogroups.com>
> Sent: Saturday, January 26, 2002 11:58 PM
> Subject: Re: [tuning] updated definition: "just intonation"
>
>
> > [me, monz]
> > I've finally added a lattice-diagram of an extensive
> > 5-limit JI system to my Tuning Dictionary entry for
> > "just intonation":
> >
> > http://www.ixpres.com/interval/dict/just.htm
>
>
> I once again object to this definition in that the greek
> theory is universally recognized as using Just intonation
> long before the idea of limits came along. To have to refer
> to them as Extended just intonation is absurd. It also ignores
> that there is already the historical use by composers about
> their own music such Partch and Harrison who refer to their
> music as Just Intonation (including myself).

These are points well taken. Thanks to your criticisms, I've
updated the Dictionary page again, and I think it's much better
now. Thanks!

http://www.ixpres.com/interval/dict/just.htm

> To try to change the terms now is just too late and it does
> nothing to further the dialog except to ghettoize the actual
> practitioners into being Extreme as in extended and out side
> of the norm. Also outside of a theoretical novelty the actual
> cultures that use 5 limit system do not and have not referred
> to it as Just intonation.
> I find the occasional impulse to change terminology already
> in existence on this list fails to recognize the limits of this
> list. It would benefit from avoiding the situation of degenerating
> into being a self proclaimed authority having say as to what
> occurs outside its boundaries

Since all of this was addressed directly to me ...

1)

We had a huge discussion on this list about the JI/rational
argument in November and December 2000. My position on this
is still exactly as I posted then, so I won't go into further
discussion of it here now. Check the list archives.

Actually, it would be very good of me to greatly "extend"
[ <groan...> pun intended ] the "just intonation" page to
include a lot of what was in that discussion, so if I don't
get around to it anytime soon, you or anyone else please give
me a nudge now and then until I do it.

2)

The main point of my post was that I had added a nice
5-limit lattice diagram that many others would hopefully
find useful.

I hadn't changed anything else in the definition (that is,
before I read your post that I'm responding to here) other
than what directly pertained to the new diagram.

3)

I don't really understand what your last paragraph means,
but in any case, the only authority I've proclaimed for
myself is as the compiler of the Tuning Dictionary.

I am under no "occasional impulse to change terminology already
in existence", but rather seek to include in the Dictionary
*all* uses of various terms by various composers and
theorists.

Indeed, a study of the history of tuning theory reveals
that theorists have cavalierly juggled terminology to mean
pretty much whatever they wanted it to -- the very recent
confusion on tuning-math over "small" and "large diesis"
is a good example.

So if you'd like to prepare an addendum to my definition
of "just intonation", then please, by all means, do.
I'll gladly include it after my own definition, as I
have with many other entries in the Dictionary with Paul
Erlich, Graham Breed, Gene Ward Smith, John Chalmers,
Margo Schulter, and many other tuning list subscribers.

Occasionally, as with "finity", I've felt the need to coin
a term for a concept in my theory that is original.
But otherwise, I'm just trying to help everyone understand
the vast and often very confusing literature on tuning
that's already been (and is still being) written.

-monz

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

> These are points well taken. Thanks to your criticisms, I've
> updated the Dictionary page again, and I think it's much better
> now. Thanks!

The way it reads now, it sounds as if you are saying three things:

(1) Ben Johnston and a few others think it only means 5-limit

(2) Everone else thinks it means rational intonation

(3) Therefore, we should follow Johnston's usage

Is it really common? What advantage does it hold over saying "5-limit"?

Hi Gene,

re:
http://www.ixpres.com/interval/dict/just.htm

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, January 27, 2002 1:17 AM
> Subject: [tuning] Re: updated definition: "just intonation"
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > These are points well taken. Thanks to your criticisms, I've
> > updated the Dictionary page again, and I think it's much better
> > now. Thanks!
>
> The way it reads now, it sounds as if you are saying three things:
>
> (1) Ben Johnston and a few others think it only means 5-limit
>
> (2) Everone else thinks it means rational intonation
>
> (3) Therefore, we should follow Johnston's usage

Huh? I state clearly on the webpage:

>> When unqualified, most writers use "just intonation" to
>> refer to a 5-limit tuning, as described in my paper.

Note that I said "most". Lots of writers on tuning theory,
for several centuries (since c. 1750 or so, IIRC ... it's in
the list archives) have used "just intonation" to refer
specifically to either the inifinite lattice of 5-limit
rational tuning, or a subset (or several) thereof.

As Kraig noted, ancient Greek theorists invented tuning systems
which used 7 and many other higher primes. But after the
Germanic invasions c. 450 AD, European theory was basically
3-limit for a thousand years. Ramos de Pareja in 1480 wrote
the first European treatise discussing a 5-limit system.

As Partch documents amusingly in _Genesis of a Music_, however,
7 was only embraced by a handful of theorists during the
1600s-1800s. Primary examples are Mersenne, Euler, and Tartini.
Right up to Schoenberg's time (and in fact right up to today)
many music-theorists do not admit 7 into their harmonic universe.

So the standard texts on harmony, while they really don't
have much to say about ratios anyway, have in general not
included 7 at all as part of the explanation of harmonic
principles.

So most of the writing about JI or rational tuning, at
least as it relates to "common-practice" Eurocentric music
and theory, uses "just intonation" to refer to the 5-limit grid.

> Is it really common? What advantage does it hold over
> saying "5-limit"?

As I said in my post to Kraig, my goals with the Tuning
Dictionary are twofold: 1) to keep up with, and provide a
base for, current developments in tuning theory, and
2) to help researchers understand the confusing body of
tuning literature that already exists.

I agree with you that using "5-limit" as a qualifier is
much better and more precise. But there are an awful lot
of writers on music-theory who haven't read Partch, don't
know about my website, and don't subscribe to any of the
tuning lists. They'll probably say "just" when they mean
"5-limit".

-monz

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

> Huh? I state clearly on the webpage:
>
> >> When unqualified, most writers use "just intonation" to
> >> refer to a 5-limit tuning, as described in my paper.

You said it, but you didn't support it--you give one example of someone who thinks so, and a great many examples of people who don't. Where is your evidence it is true?

> Note that I said "most". Lots of writers on tuning theory,
> for several centuries (since c. 1750 or so, IIRC ... it's in
> the list archives) have used "just intonation" to refer
> specifically to either the inifinite lattice of 5-limit
> rational tuning, or a subset (or several) thereof.

I don't think so--many people simply didn't talk about primes higher than five, which is different than saying primes higher than five are "extended" or don't count as just. On the other hand, those people who *did* talk about seven seem to have treated it like five.

> As Partch documents amusingly in _Genesis of a Music_, however,
> 7 was only embraced by a handful of theorists during the
> 1600s-1800s. Primary examples are Mersenne, Euler, and Tartini.
> Right up to Schoenberg's time (and in fact right up to today)
> many music-theorists do not admit 7 into their harmonic universe.

Not admitting it into your harmonic universe is not the same as saying it isn't just intonation.

> I agree with you that using "5-limit" as a qualifier is
> much better and more precise. But there are an awful lot
> of writers on music-theory who haven't read Partch, don't
> know about my website, and don't subscribe to any of the
> tuning lists. They'll probably say "just" when they mean
> "5-limit".

There may a number of people like me, who would never, ever think that "just intonation" had anything to do with the 5-limit unless that was clearly stated. Certainly, unless someone says so I am hardly going to assume that is what they mean, and it seems to me you jump to conclusions if you think not discussing seven is the same as having an opinion about it.

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, January 27, 2002 2:30 AM
> Subject: [tuning] Re: updated definition: "just intonation"
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > Huh? I state clearly on the webpage:
> >
> > >> When unqualified, most writers use "just intonation" to
> > >> refer to a 5-limit tuning, as described in my paper.
>
> You said it, but you didn't support it--you give one example
> of someone who thinks so, and a great many examples of people
> who don't. Where is your evidence it is true?

I've been working non-stop all day today updating my Dictionary
webpages, making lattices for them, clarifying and expanding
older versions, etc. Please forgive me if it takes a while
longer (may be days, may be years ... I don't get paid for this)
to put in the citations supporting what I write.
But they'll eventually be there.

I also specifically referred you to the tuning list archives
of November-December 2000, wherein you will find a very big
discussion of this topic, complete with several references
to the first occurence of the term "just intonation" in
English, German, and Italian, and many other relevant writings.

> > Note that I said "most". Lots of writers on tuning theory,
> > for several centuries (since c. 1750 or so, IIRC ... it's in
> > the list archives) have used "just intonation" to refer
> > specifically to either the inifinite lattice of 5-limit
> > rational tuning, or a subset (or several) thereof.
>
> I don't think so--many people simply didn't talk about primes
> higher than five, which is different than saying primes higher
> than five are "extended" or don't count as just. On the other hand,
> those people who *did* talk about seven seem to have treated it like five.

I'm not sure what your last sentence means.

But I'm sorry, you're simply wrong if you mean to imply that
the majority of writers on harmony "simply didn't talk about
primes higher than five". Almost every author of a harmony
textbook makes a reference to the overtone series, and notes
how "7 and 11 are out of tune, therefore excluded from the
harmonic system".

If you don't believe me, go to a big library and start looking
at one harmony text after another. You'll see.

> > As Partch documents amusingly in _Genesis of a Music_, however,
> > 7 was only embraced by a handful of theorists during the
> > 1600s-1800s. Primary examples are Mersenne, Euler, and Tartini.
> > Right up to Schoenberg's time (and in fact right up to today)
> > many music-theorists do not admit 7 into their harmonic universe.
>
> Not admitting it into your harmonic universe is not the same
> as saying it isn't just intonation.

Yes it is. The term "just intonation" refers specifically to
the so-called "purity" of intervals which sound like those
having small-integer ratios. If 7 is not admitted into the
system of ratios which describe any given author's version
of "just intonation", then can't it be logically deduced that
that author is saying that "7 isn't just intonation"?

> > I agree with you that using "5-limit" as a qualifier is
> > much better and more precise. But there are an awful lot
> > of writers on music-theory who haven't read Partch, don't
> > know about my website, and don't subscribe to any of the
> > tuning lists. They'll probably say "just" when they mean
> > "5-limit".
>
> There may a number of people like me, who would never, ever
> think that "just intonation" had anything to do with the
> 5-limit unless that was clearly stated. Certainly, unless
> someone says so I am hardly going to assume that is what
> they mean, and it seems to me you jump to conclusions if
> you think not discussing seven is the same as having an
> opinion about it.

Gene, I really don't want this to flare up into the same kind
of heated debate we had here before. You and anyone else are
perfectly free to define "just intonation" however you please,
and to include whatever prime-factors into your tunings that
you'd like to have.

But *HISTORICALLY* (is that enough emphasis?), "just intonation"
refers to 5-limit tuning. Period. That didn't really begin to
change until Parch came along, c. 1930.

Please try to understand my position on this a little better.
Barbershop Quartet style is based entirely on 7-limit adaptive
tuning, and is perhaps the finest example of 7-limit harmony
in all its sonorous beauty. And I'll be the first to call
Barbershop an example of "just intonation".

But *MOST* traditional music-theory treatises mean 5-limit
when they say "just intonation". As I explained above, most
of the authors who "don't discuss" 7 do indeed dismiss it
before eliminating from their discussion. That can't be
called not having an opinion about the matter.

I'm really sorry if you don't like that or agree with it,
but that's how it is, and it's in the past so it can't be changed.
Have patience, and eventually my definition will cite the
pertinent literature.

-monz

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I guess if you are emphasizing 5-limit as JI (and giving the poor 7
limit...and others short shrift) then you are admitting a particular bias.
You can spell that bias out in advance of the definition(s) for better
communication.

BTW, Johann Philip Kirnberger loved the 7th limit and he composed with it on
occasion (and this from Bach's most important theory student).

My preference is to consider all music microtonal with the rational numbers
considered JI (including 3-limit Pythagorean), and irrational numbers
(tempered music).

Best to all, Johnny Reinhard

Geneward Smith!
I totally agree with you on this. to Limit Just intonation to 5 limit is arbitrary and as i point out establishes a reactionary background of labeling the very composers who use it as extremist. JOE IS FAILING TO SEE HOW HE LAYS THE GROUND WORK FOR A BACKLASH.The fact also is that 5 limit just was never a practiced system in Europe and once again the cultures
that use them (who the hell cares what Europe thinks anyway since they didn't use it) have never referred it to Just Intonation.

genewardsmith wrote:

>
> You said it, but you didn't support it--you give one example of someone who thinks so, and a great many examples of people who don't. Where is your evidence it is true?
>
>

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

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

> From: Kraig Grady <kraiggrady@anaphoria.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, January 27, 2002 10:04 AM
> Subject: Re: [tuning] Re: updated definition: "just intonation"
>
>
> Geneward Smith!
> I totally agree with you on this. to Limit Just
> intonation to 5 limit is arbitrary

I'm sorry Kraig, but no, it's not arbitrary. It's a
big part of the history of tuning theory.

> and as i point out establishes a reactionary background
> of labeling the very composers who use it as extremist.
> JOE IS FAILING TO SEE HOW HE LAYS THE GROUND WORK FOR A
> BACKLASH.

Yes, and I will continue to fail to see it, because it's
a figment of *your* imagination! I am not "labeling the
very composers who use it as extremist", unless you distort
what I wrote on my webpage to make it seems as tho I'm saying
what *you* claim I'm saying. READ IT AGAIN, KRAIG!

> The fact also is that 5 limit just was never a practiced
> system in Europe and once again the cultures that use them
> (who the hell cares what Europe thinks anyway since they
> didn't use it) have never referred it to Just Intonation.

But hordes of theorists writing harmony textbooks *HAVE*
referred to 5-limit as "just intonation", and most of them
explicitly excluded 7. So, their work doesn't matter?

I really don't understand why there's such a big deal being
made about this. Sheesh, talk about pseudo-religious fundamentalism!

I've made it very clear on my webpage that what constitutes
"just intonation" is a matter of opinion, and that those opinions
diverge very widely.

If you want to write 499-limit music and call it JI, fine,
go ahead and do so. I'm basically in agreement. The name of
book, after all, is "JustMusic", and it explores historical
tuning systems with primes up to 499 (Boethius's enharmonic).

But if I *failed* to point out that the term "just intonation"
has a historical usage of at least a couple of centuries wherein
it is taken to mean essentially 5-limit JI, then I'd be doing
a tremendous disservice to all those who utilize my Dictionary
as a primary resource in their tuning education.

Again I ask, why is this such a big deal?

Let's debate stuff that's more worthy of debate, instead of
rehashing this dead horse again.

love / peace / harmony ...

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

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--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: Kraig Grady <kraiggrady@a...>

> > The fact also is that 5 limit just was never a practiced
> > system in Europe and once again the cultures that use them
> > (who the hell cares what Europe thinks anyway since they
> > didn't use it) have never referred it to Just Intonation.

This is a non-issue because "Just Intonation" and its accepted
translations are in European languages, so once one goes
outside Europe, there's no anyone *could* have referred to
anything as "Just Intonation". Perhaps only Europe is capable of
producing such an ambiguous, poorly defined term -- which
many authors claim has a different definition depending on
whether the "J" and the "I" are capitalized or not!

Joe!
BTW JI also doesn't have to have a 1/1
And this theory was centuries old before it was imported to Europe and are not homegrown
inventions of europe. 5-limit JI has no historical practice and served as a theoretical excuse for
meantone
The very concept they got from the Greeks and Persians and they used 11 and 13s
and what ever ratio they wished. JI is nothing more than the use of ratios to generate scales.
There are all types of books that imply that the earth is the center of the universe as a
part or astronomy's history yet no one bothers to give them any credence now.

fundamentalism?
you are the one "defining" things like the council of trent
monz wrote:

>
> I'm sorry Kraig, but no, it's not arbitrary. It's a
> big part of the history of tuning theory.
>

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

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

Hello ALL, it seems that our widely accepted dear old 5-limit JI in
Indian music may tremble a little.

Is the following statement true? :
JI is an intonation in which octave doubling cannot be achieved.

Regards,
Haresh.

Paul!
And these translations are filled with mistakes. Boetius believed that the mode went from low
to high. Just because it is in a book means that it is right or even of valid opinion.
5 limit music doesn't exist in Europe so it is hard to conceive of these things as little more
that numerical metaphysics. The omission of 7 was influenced by the church so it is possible that
those who wrote the books had to do so due to the political pressure.
Literature under such oppression can not be considered a reliable source of the view point of
the authors. Why should we consider it any more than those astronomy books that tell us that the
earth was the center of the universe.?

paulerlich wrote:

> This is a non-issue because "Just Intonation" and its accepted
> translations are in European languages, so once one goes
> outside Europe, there's no anyone *could* have referred to
> anything as "Just Intonation". Perhaps only Europe is capable of
> producing such an ambiguous, poorly defined term -- which
> many authors claim has a different definition depending on
> whether the "J" and the "I" are capitalized or not!
>
>

-- 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..., "hbakshi1" <hareshbakshi@h...> wrote:
> Hello ALL, it seems that our widely accepted dear old 5-limit
JI in
> Indian music may tremble a little.
>
> Is the following statement true? :
> JI is an intonation in which octave doubling cannot be
achieved.

Not true at all. Why would you think so?

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> Paul!
> And these translations are filled with mistakes. Boetius
believed that the mode went from low
> to high. Just because it is in a book means that it is right or
even of valid opinion.

I agree 100%!

> The omission of 7 was influenced by the church

I've never heard this one before. Would you elaborate, please?
Certainly seems, plasible, considering the contemperaneous
Galileo connection suggested by the below . . .

> so it is possible that
> those who wrote the books had to do so due to the political
pressure.
> Literature under such oppression can not be considered a
reliable source of the view point of
> the authors. Why should we consider it any more than those
astronomy books that tell us that the
> earth was the center of the universe.?

Paul
In reference to the tritone. you can't use 7s without them popping up.
(pardon the pun)

paulerlich wrote:

>
> > The omission of 7 was influenced by the church
>
> I've never heard this one before. Would you elaborate, please?
> Certainly seems, plasible, considering the contemperaneous
> Galileo connection suggested by the below . . .
>
>

-- 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..., "paulerlich" <paul@s...> wrote:
> --- In tuning@y..., "hbakshi1" <hareshbakshi@h...> wrote:
> > Hello ALL, it seems that our widely accepted dear old 5-limit
> JI in
> > Indian music may tremble a little.
> >
> > Is the following statement true? :
> > JI is an intonation in which octave doubling cannot be
> achieved.
>
> Not true at all. Why would you think so? >>>>>>>>>>

Hello Paul,

Because, is it not true that we can take the "spiral" to infinity
without ever reaching 2:1 ?

Haresh.

--- In tuning@y..., "hbakshi1" <hareshbakshi@h...> wrote:

> > > Is the following statement true? :
> > > JI is an intonation in which octave doubling cannot be
> > achieved.
> >
> > Not true at all. Why would you think so? >>>>>>>>>>
>
> Hello Paul,
>
> Because, is it not true that we can take the "spiral" to infinity
> without ever reaching 2:1 ?
>
> Haresh.

JI is an infinite lattice based on rational intervals, and 2:1 is a
rational interval that is very seldom omitted by those using JI.

A "spiral" of fifths, as 3:2 fifths, "nearly misses" the 7-octave
mark, that is, 128:1, so 2:1 doesn't really come into play anyway ,
unless you posit it to begin with, as in Pythagorean tuning.

--- In tuning@y..., "hbakshi1" <hareshbakshi@h...> wrote:

> Because, is it not true that we can take the "spiral" to infinity
> without ever reaching 2:1 ?

Actually it is the 1:1 that one never reaches. Remember, the
notes in the "spiral of fifths" are octave-reduced to begin with, i.e.,
2:1 is already assumed, and it is the 1:1 that becomes the focus
of attention -- how close does one come to it after 12, 53, or 665
fifths, or what have you.

> From: Kraig Grady <kraiggrady@anaphoria.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, January 27, 2002 2:33 PM
> Subject: Re: [tuning] Re: updated definition: "just intonation"
>
>
> Joe!
> BTW JI also doesn't have to have a 1/1

Of course. Hmmm, I should definitely add that to the definition.
Thanks.

> And this theory was centuries old before it was imported
> to Europe and are not homegrown inventions of europe.

And I've never made any claims to the contrary. But please note,
as Paul wrote, that the term itself ("just intonation" and its
equivalents in German, French, Italian, etc.) is European and
therefore doesn't even apply in non-European musical cultures.

Just because Europeans didn't invent it doesn't mean they
didn't write plenty about it.

> 5-limit JI has no historical practice and served as a
> theoretical excuse for meantone

Woah! I think you're overstating the case at least a little.
Any Renaissance _a cappella_ performance certainly had a
foundation in 5-limit JI, even if the actual presentation
was more likely to be adaptive-JI or adaptive tuning.

Yes, JI did serve "as a theoretical excuse for meantone",
altho I don't think I'd put it that negatively. Meantone
has its own unique properties that were explored
enthusiastically by European and Eurocentric composers.

The JI basis, as far as I can tell, is limited pretty
much to considerations concerning the "major" and
"minor 3rds" and their inversions, and perhaps a few
other intervals that imply intervals which are close to
1:1 on the JI lattice.

But even by the time of Beethoven, both the JI and meantone
basis was waning and being slowly replaced by one which
took 12-EDO as its starting point, and this was a process
which reached its culmination with Schoenberg and the academic
legacy which flowed from his work, and then also with bebop jazz.

> The very concept they got from the Greeks and Persians
> and they used 11 and 13s and what ever ratio they wished.

I'd really like to know more about the Persian background
on European music-theory. You've mentioned it several times
in the past, and unfortunately I don't know much about Persian
music. Please post as much about this as you can, or give
some references.

> JI is nothing more than the use of ratios to generate scales.

Ah ... but what about that Hammond Organ tuning? That's
generated entirely from ratios, but sounds exactly like 12-EDO.
So ... is it 12-EDO, or JI, or something else? If the latter,
then what? What would *you* call it?

Or for another example, how about my "rational Canasta" tuning?
I wrote:

> Message 23129
> From: "monz" <joemonz@y...>
> Date: Fri May 18, 2001 3:13 pm
> Subject: Ratios for MIRACLEs (was: Re: second Monzo Canasta Ztar mapping)
/tuning/topicId_22793.html#23129
>
> ...
>
> So here's a challenge: can anyone come up with 13-limit
> rational pitch sets which will approximate Blackjack and
> Canasta to good advantage? Then you can use my software
> to put these scales on your computer keyboard, and play away!
>
> (Hmmm... that's pretty weird... looking for a rational
> approximation for subsets of an equal-temperament which
> itself is supposed to be good at approximating ratios...
> I see Escher pictures in my mind...)

And then in message 23188 I proceeded to present a rational
tuning which is audibly indistinguishable from the 72-EDO
version of the Canasta tuning:
/tuning/topicId_22793.html#23188

But please note: these ratios have nothing to do with the
simple ratios which are implied by Canasta. They were
meant to be a rational approximation to 72-EDO. Why did
I bother to do that? Because my goal was to map Canasta
to the computer keyboard for use in my JustMusic software,
and JustMusic so far can only deal with rational pitches.

So what about my Rational Canasta tuning? Is it JI? Or not?

> There are all types of books that imply that the earth
> is the center of the universe as a part or astronomy's history
> yet no one bothers to give them any credence now.

Well, "no-one bothers to give them any credence now" if their
consideration is currently astronomical data or research.
But what if one's goal is to understand the medieval European
mind-set? Then if you see a book that says that the earth
is the center of the universe, I think you'd *better* give
it some credence, or you won't have any idea how those people
thought about the universe or their place in it. Same thing
goes for what's already been written about JI. B A M !!

> fundamentalism?
> you are the one "defining" things like the council of trent

OK, you brought this up, not me ... I'm going to quote only
the very beginning of the Council of Trent (because the darn
thing is so long):

http://history.hanover.edu/texts/trent/ctbull.html

>> THE BULL OF INDICTION
>> OF THE SACRED OECUMENICAL AND GENERAL COUNCIL OF TRENT
>> UNDER THE SOVEREIGN PONTIFF, PAUL III
>>
>>
>> PAUL, bishop, servant of the servants of God, for the
future memory hereof.
>>
>>
>> At the beginning of this our pontificate, -- which, not
>> for any merits of our own, but of its own great goodness,
>> the providence of Almighty God hath committed unto us,
>> -- already perceiving unto what troubled times, and unto
>> how many embarrassments in almost all our affairs, our pastoral
>> solicitude and watchfulness were called; we would fain indeed
>> have remedied the evils wherewith the Christian commonweal had
>> been long afflicted, and well-nigh overwhelmed; but we too,
>> as men compassed with infirmity, felt our strength unequal
>> to take upon us so heavy a burthen. For, whereas we saw that
>> peace was needful to free and preserve the commonweal from
>> the many impending dangers, we found all replete with enmities
>> and dissensions; and, above all, the (two) princes, to whom
>> God has entrusted well-nigh the whole direction of events,
>> at enmity with each other. Whereas we deemed it necessary
>> that there should be one fold and one shepherd, for the Lord's
>> flock in order to [Page 2] maintain the Christian religion
>> in its integrity, and to confirm within us the hope of heavenly
>> things; the unity of the Christian name was rent and well-nigh
>> torn asunder by schisms, dissensions, heresies.

Now I really don't think that my Tuning Dictionary has anything
at all to do with that, excepting that I *am* all for peace
and harmony (enjoy the pun if you'd like).

All I'm trying to do is assemble a reference that will make it
easier for people who are interested in reading the huge literature
on tuning.

If you don't like my definition, fine. Like I said, write your
own, and I'll include it. Actually, I'd *prefer* to include your
dissenting views rather than leave them out. So, stop with the
knee-jerk-response posts, and send me a real definition of JI
as you know it to be, and it will forever become a part of the
Online Dictionary of Tuning Terms.

Now ... I *REALLY* don't want to write any more about this !!!!

-monz

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Kraig Grady schrieb:
>
> Paul
> In reference to the tritone. you can't use 7s without them popping up.

The very term hints at a Pythagorean system. They might not
have looked any further than 3-limit because of the trinity,
but apart from that...

And 3-limit tunings certainly have been in use before The
Church started to codify its music.

A related issue: it seems very very likely to me that much
of the flexibility of vocal music may have disappeared
because of more or less inflexible instruments that were
used either for accompaniment or, perhaps, as a measuring
tool for codification. Using string harmonics to tune up
psalteries and such. Non-Pythagorean intervals were not
measured and outlawed, they didn't fit the system and were
discarded.

klaus

Klaus!
I will tend to agree that that the tritone as it was "outlawed " was originally thought of in
the context of 3 or 5 limit music. I think though anyone proposing the use of 7 limit and to
state that the 7/5 was a consonance would have led (an assumption) to that favorite form of
weekend entertainment - burning in the public square.

klaus schmirler wrote:

> Kraig Grady schrieb:
> >
> > Paul
> > In reference to the tritone. you can't use 7s without them popping up.
>
> The very term hints at a Pythagorean system. They might not
> have looked any further than 3-limit because of the trinity,
> but apart from that...
>
> And 3-limit tunings certainly have been in use before The
> Church started to codify its music.
>
> A related issue: it seems very very likely to me that much
> of the flexibility of vocal music may have disappeared
> because of more or less inflexible instruments that were
> used either for accompaniment or, perhaps, as a measuring
> tool for codification. Using string harmonics to tune up
> psalteries and such. Non-Pythagorean intervals were not
> measured and outlawed, they didn't fit the system and were
> discarded.
>
> klaus
>

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

> Woah! I think you're overstating the case at least a little.
> Any Renaissance _a cappella_ performance certainly had a
> foundation in 5-limit JI, even if the actual presentation
> was more likely to be adaptive-JI or adaptive tuning.

Right -- so any assignment of pitches to fixed ratios would have
been off the mark.
>
> The JI basis, as far as I can tell, is limited pretty
> much to considerations concerning the "major" and
> "minor 3rds" and their inversions, and perhaps a few
> other intervals that imply intervals which are close to
> 1:1 on the JI lattice.

Like 4:3 and 3:2.
>
> Or for another example, how about my "rational Canasta"
tuning?

Monz, while you were away, we made a great deal of progress in
designing rational MIRACLE tunings -- the "broken
consonances", at least in 7-limit, can be made to all fall into one
of three categories -- errors of 0.7 cents, 7.0 cents, and 7.7 cents
-- none of which are _that_ egregious. Check up on this at
tuning-math.

Kraig Grady schrieb:
>
> Klaus!
> I will tend to agree that that the tritone as it was "outlawed " was originally thought of in
> the context of 3 or 5 limit music.

tritone=3 toni
the tonus maior being 9/8

Or, arguing from the side of the issue, if music using the
7/5 existed, it certainly wasn't outlawed after someone took
the trouble to measure the interval. If so, some edict would
be likely to mention it.

BTW: Are you thinking of the renaissance or thereabouts when
5-limit music was actually heard of in a liturgical context?
I wouldn't know that I would argue differently, but anyway I
was thinking of monophonic chants and the codification of
part of the repertoire (across Europe) by Gregory I. And the
possiblity that other parts of the repertoire were
extinguished and that this didn't happen in the Byzantian
empire.

klaus

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> Woah! I think you're overstating the case at least a little.
> Any Renaissance _a cappella_ performance certainly had a
> foundation in 5-limit JI, even if the actual presentation
> was more likely to be adaptive-JI or adaptive tuning.

Why not say instead it had a foundation in meantone?

> Ah ... but what about that Hammond Organ tuning? That's
> generated entirely from ratios, but sounds exactly like 12-EDO.
> So ... is it 12-EDO, or JI, or something else? If the latter,
> then what? What would *you* call it?

If it's close enough to 12-et, why not call it that?

Klaus !
I thought all this happened around Palestrina.

Lets ask Margo
Margo it safe to come out now!

klaus schmirler wrote:

>
> Or, arguing from the side of the issue, if music using the
> 7/5 existed, it certainly wasn't outlawed after someone took
> the trouble to measure the interval. If so, some edict would
> be likely to mention it.
>

-- 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..., Kraig Grady <kraiggrady@a...> wrote:

> I totally agree with you on this. to Limit Just intonation to 5 limit is arbitrary and as i point out establishes a reactionary background of labeling the very composers who use it as extremist.

I thought saying Harrison and Partch, et al "insisted" on calling what they did JI was a little strange.

>JOE IS FAILING TO SEE HOW HE LAYS THE GROUND WORK FOR A BACKLASH.

I'm more concerned that he is laying the groundwork for more confusion. I don't see the evidence that JI = 5-limit in the discussions of contemporary theorists, as evidenced most easily by simply reading this list. Joe suggests that most harmony books have some confused statement somewhere about the overtone series being out of tune, which I haven't noticed, but I am no expert on these things. I do recall statements that it is out of accord with 12-et, but that does not support Joe's position.

However, even if they do say this, are they doing so in the context of attempting to define JI? It seems to me more likely that they are preparing to say they are justified in ignoring everything but 12-et.
My experience of theory books is that if they discuss tuning at all they mostly assume 12 notes to the octave was given by God on Mt. Sinai, and if they attempt to discuss why, they fail to make sense.

> From: genewardsmith <genewardsmith@juno.com>
> To: <tuning@yahoogroups.com>
> Sent: Sunday, January 27, 2002 4:38 PM
> Subject: [tuning] Re: updated definition: "just intonation"
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > Woah! I think you're overstating the case at least a little.
> > Any Renaissance _a capella_ performance certainly had a
> > foundation in 5-limit JI, even if the actual presentation
> > was more likely to be adaptive-JI or adaptive tuning.
>
> Why not say instead it had a foundation in meantone?

Not for _a capella_ vocal music.

Meantone was originally created for keyboards, around 1520,
then extended to all the other instruments, which in fact
retained it during the 1700s after keyboards had generally
been switched to well-temperament.

When people sing without the support of any instruments
-- good singers, that is -- their natural inclination is
to tune harmonies according to something resembling
low-integer JI, generally 5- or 7-limit. That singing
is generally not *in* JI is because of the typical commatic
problems. So singers intuitively work out some form of
adaptive tuning, trying to keep the vertical sonorities
more-or-less close to subsets of the harmonic series.

Meantone only enters the singing picture if the singers
are being accompanied by keyboards tuned in meantone.

Certainly, the notation and harmonic practice of Renaissance
vocal music is grounded in meantone to some extent. But
without being forced to preserve the meantone pitches,
singers will adjust the intonation slightly to achieve
greater consonance on sustained tones.

> > Ah ... but what about that Hammond Organ tuning? That's
> > generated entirely from ratios, but sounds exactly like 12-EDO.
> > So ... is it 12-EDO, or JI, or something else? If the latter,
> > then what? What would *you* call it?
>
> If it's close enough to 12-et, why not call it that?

My point exactly. I'd call this particular example an RI
(rational intonation) which emulates 12-EDO. But it certainly
is no form of JI.

-monz

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--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > Woah! I think you're overstating the case at least a little.
> > Any Renaissance _a cappella_ performance certainly had a
> > foundation in 5-limit JI, even if the actual presentation
> > was more likely to be adaptive-JI or adaptive tuning.
>
> Why not say instead it had a foundation in meantone?

Good idea!

>
> > Ah ... but what about that Hammond Organ tuning? That's
> > generated entirely from ratios, but sounds exactly like 12-EDO.
> > So ... is it 12-EDO, or JI, or something else? If the latter,
> > then what? What would *you* call it?
>
> If it's close enough to 12-et, why not call it that?

Because you upset the people who want to keep 12-tET and JI mutually
exclusive terms.

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: genewardsmith <genewardsmith@j...>
> > To: <tuning@y...>
> > Sent: Sunday, January 27, 2002 4:38 PM
> > Subject: [tuning] Re: updated definition: "just intonation"
> >
> >
> > --- In tuning@y..., "monz" <joemonz@y...> wrote:
> > >
> > > Woah! I think you're overstating the case at least a little.
> > > Any Renaissance _a capella_ performance certainly had a
> > > foundation in 5-limit JI, even if the actual presentation
> > > was more likely to be adaptive-JI or adaptive tuning.
> >
> > Why not say instead it had a foundation in meantone?
>
>
> Not for _a capella_ vocal music.

Adaptive-JI and adaptive tuning, in the repertoire in question, would
have produced tunings that track meantone very closely. This is only
one reason one might sensibly argue "it had a foundation in meantone".

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:

> I'm more concerned that he is laying the groundwork for more
>confusion. I don't see the evidence that JI = 5-limit in the
>discussions of contemporary theorists, as evidenced most easily by
>simply reading this list. Joe suggests that most harmony books have
>some confused statement somewhere about the overtone series being
>out of tune, which I haven't noticed, but I am no expert on these
>things. I do recall statements that it is out of accord with 12-et,
>but that does not support Joe's position.
>
> However, even if they do say this, are they doing so in the context
>of attempting to define JI? It seems to me more likely that they are
>preparing to say they are justified in ignoring everything but 12-et.

No offense Gene, but you're clearly not too familiar with the
relevant literature here, which would _not_ be the literature that
attempts to justify 12-tET. Have you read Benedetti? Rameau? Bruckner?

>My experience of theory books is that if they discuss tuning at all
>they mostly assume 12 notes to the octave was given by God on Mt.
>Sinai, and if they attempt to discuss why, they fail to make sense.

This is mainly a phenomenon of books of the last century. You should
look at English-language texts from the 19th century -- they
understood that triadic diatonic tuning was "based" on meantone in
the sense you mentioned earlier -- very few musicians have any
concept of this today.

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> Joe!
> BTW JI also doesn't have to have a 1/1
> And this theory was centuries old before it was imported to
Europe and are not homegrown
> inventions of europe.

Perhaps, then, we should look to the original, non-European
terms used to express what you're calling "just intonation", and
decide whether a better etymology might be culled . . . of course,
if there's no written record, we're at a bit of a disadvantage . . .

>5-limit JI has no historical practice and served as a theoretical
>excuse for
> meantone

Amen, Kraig. 99% true. The only exception may be the keyboard
tuning from 1420-1480, schismic in concept but no doubt tuned
in
"JI" (in the key of D major) as well as could be ascertained with
the tools of the time.

> fundamentalism?
> you are the one "defining" things like the council of trent

Kraig, have you even looked at Monz' definition pages in
question?

Monz, take a look at this:

/tuning/files/perlich/scales/blackjust4.g
if

I've shown all the "broken consonances" that occur in this
detempering of Blackjack -- each of them is off by

225:224 = 7.7 cents,
2401:2400 = 0.7 cents, or
16875:16807 = 7.0 cents

Hopefully, the diagram is self-explanatory.

The same think could be done for Canasta, if you wished . . .

Thanks to a contribution from a colleague who wishes
to remain anonymous, I've made a huge update to the
"just intonation" definition, which now quotes the entire
_New Grove's Dictionary of Music and Musicians_ entry
by Mark Lindley, including an extensive bibliography
of treatises and papers concerning JI.

http://www.ixpres.com/interval/dict/just.htm

love / peace / harmony ...

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

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> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, January 28, 2002 6:03 PM
> Subject: [tuning] For Monz -- centered 7-limit JI version of Blackjack
>
>
> Monz, take a look at this:
>
> /tuning/files/perlich/scales/blackjust4.g
> if
>
> I've shown all the "broken consonances" that occur in this
> detempering of Blackjack -- each of them is off by
>
> 225:224 = 7.7 cents,
> 2401:2400 = 0.7 cents, or
> 16875:16807 = 7.0 cents

This is awsome, Paul! Thanks!

> Hopefully, the diagram is self-explanatory.

So in other words, this actually *is* JI? And the
"broken consonances", whose connections you've colored
in grey, are not exactly what they appear to be?

> The same think could be done for Canasta, if you wished . . .

Please do! "Canasta" is the MIRACLE tuning I personally
wish to start exploring.

... unless the extra notes in "Studloco" seem worth it
(because I just *love* that name!).

-monz

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--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
>
>
> Thanks to a contribution from a colleague who wishes
> to remain anonymous, I've made a huge update to the
> "just intonation" definition, which now quotes the entire
> _New Grove's Dictionary of Music and Musicians_ entry
> by Mark Lindley, including an extensive bibliography
> of treatises and papers concerning JI.
>
> http://www.ixpres.com/interval/dict/just.htm

A 5-second glance revealed one error:

"Zarlino's mystical doctrine of the senario (the doctrine that the
numbers 16 are the essence of music)"

That should be 1-6 rather than 16.

I'll take a closer look when I get a chance.

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > From: paulerlich <paul@s...>
> > To: <tuning@y...>
> > Sent: Monday, January 28, 2002 6:03 PM
> > Subject: [tuning] For Monz -- centered 7-limit JI version of
Blackjack
> >
> >
> > Monz, take a look at this:
> >
> >
/tuning/files/perlich/scales/blackjust4.g
> > if
> >
> > I've shown all the "broken consonances" that occur in this
> > detempering of Blackjack -- each of them is off by
> >
> > 225:224 = 7.7 cents,
> > 2401:2400 = 0.7 cents, or
> > 16875:16807 = 7.0 cents
>
> This is awsome, Paul! Thanks!
>
> > Hopefully, the diagram is self-explanatory.
>
> So in other words, this actually *is* JI?

Yes -- in the "ratio" definition.

> And the
> "broken consonances", whose connections you've colored
> in grey, are not exactly what they appear to be?

They're out of tune by the indicated interval.
>
> > The same think could be done for Canasta, if you wished . . .
>
>
> Please do! "Canasta" is the MIRACLE tuning I personally
> wish to start exploring.

Perhaps Gene could generate the corresponding ratios more quickly
than I could.

Also, look at the other "Blackjust" gifs in that directory. You'll
see other, non-centered possibilities, giving more just tetrads or
hexanies.

> From: paulerlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Tuesday, January 29, 2002 12:35 PM
> Subject: [tuning] Re: updated definition: "just intonation"
>
>
> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> >
> > ... huge update to the "just intonation" definition, which
> > now quotes the entire _New Grove's Dictionary of Music and
> > Musicians_ entry by Mark Lindley, ...
> >
> > http://www.ixpres.com/interval/dict/just.htm
>
> A 5-second glance revealed one error:
>
> "Zarlino's mystical doctrine of the senario (the doctrine that the
> numbers 16 are the essence of music)"
>
> That should be 1-6 rather than 16.
>
> I'll take a closer look when I get a chance.

Wow, thanks, Paul! Glad you caught that, it's a big error.

-monz

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

> > Please do! "Canasta" is the MIRACLE tuning I personally
> > wish to start exploring.

> Perhaps Gene could generate the corresponding ratios more quickly
> than I could.

There are, of course, many ways to detemper Canasta. Here is something based on a "notation" <81/80,100/99,225/224,243/242,385/384>

[1, 55/54, 135/128, 16/15, 12/11, 9/8, 55/48, 64/55, 6/5, 11/9, 5/4, 165/128, 72/55, 4/3, 11/8, 45/32, 64/45, 16/11, 3/2, 55/36, 256/165, 8/5, 18/11, 5/3, 55/32, 96/55, 16/9, 11/6, 15/8, 256/135, 108/55]

Here is a Studloco block, from the same starting point:

[1, 45/44, 25/24, 288/275, 16/15, 12/11, 10/9, 9/8, 55/48, 64/55, 32/27, 6/5, 11/9, 5/4, 225/176, 32/25, 72/55, 4/3, 15/11, 11/8, 45/32, 64/45, 16/11, 22/15, 3/2, 55/36, 25/16, 352/225, 8/5, 18/11, 5/3, 27/16, 55/32, 96/55, 16/9, 9/5, 11/6, 15/8, 275/144, 48/25, 88/45]

In-Reply-To: <a38g8e+hhit@eGroups.com>
genewardsmith wrote:

> Here is a Studloco block, from the same starting point:
>
> [1, 45/44, 25/24, 288/275, 16/15, 12/11, 10/9, 9/8, 55/48, 64/55,
> 32/27, 6/5, 11/9, 5/4, 225/176, 32/25, 72/55, 4/3, 15/11, 11/8, 45/32,
> 64/45, 16/11, 22/15, 3/2, 55/36, 25/16, 352/225, 8/5, 18/11, 5/3,
> 27/16, 55/32, 96/55, 16/9, 9/5, 11/6, 15/8, 275/144, 48/25, 88/45]

How about

49/48 25/24 21/20 16/15 12/11 10/9 9/8 8/7 7/6 32/27 6/5 11/9 5/4 14/11
9/7 21/16 4/3 15/11 11/8 7/5 10/7 16/11 22/15 3/2 32/21 14/9 11/7 8/5
18/11 5/3 27/16 12/7 7/4 16/9 9/5 11/6 15/8 40/21 48/25 96/49 2/1

Graham

--- In tuning@y..., "genewardsmith" <genewardsmith@j...> wrote:
> --- In tuning@y..., "paulerlich" <paul@s...> wrote:
>
> > > Please do! "Canasta" is the MIRACLE tuning I personally
> > > wish to start exploring.
>
> > Perhaps Gene could generate the corresponding ratios more quickly
> > than I could.
>
> There are, of course, many ways to detemper Canasta.

We are particularly interested in making sure that none of
the "broken consonances" within the 7-limit (or if possible, the 11-
limit) are off by more than about 7.7 cents. Is this true for the
scales below?

> Here is something based on a "notation"
><81/80,100/99,225/224,243/242,385/384>
>
> [1, 55/54, 135/128, 16/15, 12/11, 9/8, 55/48, 64/55, 6/5, 11/9,
5/4, 165/128, 72/55, 4/3, 11/8, 45/32, 64/45, 16/11, 3/2, 55/36,
256/165, 8/5, 18/11, 5/3, 55/32, 96/55, 16/9, 11/6, 15/8, 256/135,
108/55]
>
> Here is a Studloco block, from the same starting point:
>
> [1, 45/44, 25/24, 288/275, 16/15, 12/11, 10/9, 9/8, 55/48, 64/55,
32/27, 6/5, 11/9, 5/4, 225/176, 32/25, 72/55, 4/3, 15/11, 11/8,
45/32, 64/45, 16/11, 22/15, 3/2, 55/36, 25/16, 352/225, 8/5, 18/11,
5/3, 27/16, 55/32, 96/55, 16/9, 9/5, 11/6, 15/8, 275/144, 48/25,
88/45]

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> Paul

> In reference to the tritone. you can't use 7s without them
popping up.
> (pardon the pun)

True, but as far as I've read, the objection from the church was
simply to the tritone (occuring as it does as about 612 cents in
Pythagorean or anything like it) and never to the explicit use of 7,
as least not once 5 was accepted. I'd be interested in evidence
to the contrary.

Also, missing the pun . . . ?

Hope you are well, Kraig . . .