question on Ben Johnston notation

I have a couple of questions on Ben Johnston notation...

It seems he is raising or lowering pitches by various commas in order
to get just intonation. There's a 33/32 higher, 32/33 lower, 25/24
higher, 24/25 lower, an 81/80 higher and a 80/81 lower, and a 35/36
higher and a 36/35 lower.

OK... so how does a performer know when they have reached that comma
"adjustment?" Is it the absence of beats??

Additionally, how does a COMPOSER know which commas go where? Are
there charts someplace of such? Or is it something I'm supposed to
already know??

I think I need a little assistance with this...

Thanks!

________ _____ _____ ____
Joseph Pehrson

> I have a couple of questions on Ben Johnston notation...

Sure . . . note, though, that we seem to have reached a consensus (Daniel
Wolf, Monz, me . . .) that it's not a great way of notating JI . . . better
would be the "standard" way where the unaltered notes form a Pythagorean
chain . . . Daniel Wolf had published (?) such an improved version of
Johnston's notation . . . but either way . . .

>OK... so how does a performer know when they have reached that comma
>"adjustment?" Is it the absence of beats??

Not necessarily! Depends on the chords written. And in Ben Johnston's
notation, even the interval D-A with no adjustments beats like a
hummingbird's wings.

>Additionally, how does a COMPOSER know which commas go >where? Are
>there charts someplace of such? Or is it something I'm >supposed to
>already know??

You mean one of two things . . . either you need to know the details of Ben
Johnston's notation system, or you need to know how to compose in JI . . .
which do you mean?

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_20929.html#20930

>
> >Additionally, how does a COMPOSER know which commas go >where?
Are there charts someplace of such? Or is it something I'm
>supposed
to already know??
>
> You mean one of two things . . . either you need to know the
details of Ben Johnston's notation system, or you need to know how to
compose in JI . . . which do you mean?

Ummm, Gee! I guess I kind of assumed that if I were using Ben
Johnston's notation I WOULD be composing in Just Intonation... but it
sounds as though this is not necessarily the case!? :(

By the way, Paul... whatever happened to your paper for the
MicroFest...?? I looked for it everyplace and couldn't find it... I
hope Brian McLaren didn't eat it...

_________ ______ _____ _
Joseph Pehrson

Hi Joseph!

>By the way, Paul... whatever happened to your paper for the
>MicroFest...?? I looked for it everyplace and couldn't find it... I
>hope Brian McLaren didn't eat it...

Nope . . . they arrived at Bill Alves' office Friday or Saturday, and they
lock his office for the weekend. :( He's sending them back to me . . . I'll
make them available for $5 . . . Ironically, the cover figure is a perfect
illustration of Ben Johnston's notation (at least in the 5-limit)!

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_20929.html#20929

> I have a couple of questions on Ben Johnston notation...

Once again, Joe, I refer you to my book (which you have).
I devote a whole chapter to Johnston's notation and theory.

The pitches in his music are generally pretty easy to tune,
because he composes entirely in just-intonation and therefore
all pitches in a simultaneity bear a clearly audible relationship
to each other.

Understanding the notation, however, is another matter.

As Paul mentioned, I and others feel that the best method is
to use a notation where the letter-names (with or without sharps
and flats) are used strictly to represent a Pythagorean system,
and other accidentals are used to indicate all prime-factors
higher than 3.

Johnston's notation is problemmatic (IMO) because it assumes
the standard 5-limit JI major scale as its basis, and then
uses all other accidentals to represent higher primes. This
has the result that anyone wishing to decipher the pitch of
a given note has to go thru a 2-step process of first determining
what is the basic scale, then determining the relationship
that note has to that basic scale.

The notation by Daniel Wolf appeared in a letter in _1/1_.
Except for the different typographical symbols for the various
accidentals, it's essentially identical to the system I propose
in my book and in this article:
http://www.ixpres.com/interval/monzo/article/article.htm

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

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_20929.html#20930

> > I have a couple of questions on Ben Johnston notation...
>
> Sure . . . note, though, that we seem to have reached a consensus
(Daniel Wolf, Monz, me . . .) that it's not a great way of notating
JI . . . better would be the "standard" way where the unaltered notes
form a Pythagorean chain . . . Daniel Wolf had published (?) such an
improved version of Johnston's notation . . . but either way . . .
>

I would be very interested in seeing Daniel Wolf's "improvement..." I
hope he will post it! Generally, in the past, he has come up with
quite logical methods of microtonal notation...

I didn't realize that the BASIS for the Ben Johnston notation was
alterations from 12-tET... I guess that would make sense from what I
was shown.

Well, that has to be a certain "flaw" in that system... correct? I
wonder why Johnston didn't think about that. Perhaps he was just
trying to be "practical?" Dunno.

> >OK... so how does a performer know when they have reached that
comma "adjustment?" Is it the absence of beats??
>
> Not necessarily! Depends on the chords written. And in Ben
Johnston's notation, even the interval D-A with no adjustments beats
like a hummingbird's wings.
>

I see... and that's, of course, 12-tET again...

> >Additionally, how does a COMPOSER know which commas go >where?
Are there charts someplace of such? Or is it something I'm >supposed
to already know??
>
> You mean one of two things . . . either you need to know the
details of Ben Johnston's notation system, or you need to know how to
compose in JI . . .
> which do you mean?

Well, I guess more the latter. I sent a message to David Doty and he
wrote a nice message back... Essentially he feels that Ben Johnston's
system (flawed or not) is not really difficult to learn, so was I was
*really* asking was how to fluently compose in JI...

Doty claims no one has written a thorough book on that subject...

Any takers?

_______ _____ ______ _____
Joseph Pehrson

Hi Joseph . . .

>I would be very interested in seeing Daniel Wolf's "improvement..." I
>hope he will post it!

I'm sure he's posted it and/or published it, and I don't think he's
following the list anymore . . . in any case, I think it's the same as
Johnston's except that the unaltered notes (A,B,C,D,E,F,G) come from
Pythagorean tuning rather than a "JI Major" conception.

>I didn't realize that the BASIS for the Ben Johnston notation was
>alterations from 12-tET...

That's not correct. It's alterations from the "JI Major" scale.

>> Not necessarily! Depends on the chords written. And in Ben
>Johnston's notation, even the interval D-A with no adjustments beats
>like a hummingbird's wings.
>>

>I see... and that's, of course, 12-tET again...

Nope . . . D-A in 12-tET is only 2 cents off a 3:2 . . . D-A in Ben
Johnston's notation is 21.5 cents off a 3:2.

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

/tuning/topicId_20929.html#21054

>
> --- In tuning@y..., jpehrson@r... wrote:
>
> /tuning/topicId_20929.html#20929
>
> > I have a couple of questions on Ben Johnston notation...
>
>
> Once again, Joe, I refer you to my book (which you have).
> I devote a whole chapter to Johnston's notation and theory.
>

Thanks, Monz for the reference!

Yes, I checked your book here... however, the "chapter" you mention
is only 4 pages in length!

> The pitches in his music are generally pretty easy to tune,
> because he composes entirely in just-intonation and therefore
> all pitches in a simultaneity bear a clearly audible relationship
> to each other.
>
> Understanding the notation, however, is another matter.
>
> As Paul mentioned, I and others feel that the best method is
> to use a notation where the letter-names (with or without sharps
> and flats) are used strictly to represent a Pythagorean system,
> and other accidentals are used to indicate all prime-factors
> higher than 3.
>
> Johnston's notation is problemmatic (IMO) because it assumes
> the standard 5-limit JI major scale as its basis, and then
> uses all other accidentals to represent higher primes. This
> has the result that anyone wishing to decipher the pitch of
> a given note has to go thru a 2-step process of first determining
> what is the basic scale, then determining the relationship
> that note has to that basic scale.
>

I think see what you're saying... In other words, making adjustments
from a 3-limit system is a more "direct" process than starting with
5-limit...especially since our standard staff is derived from a basic
3-limit system with Pythagorean...

Am I "getting the drift" on that??

> The notation by Daniel Wolf appeared in a letter in _1/1_.

I'd better hunt around for this... Do you remember the issue
number.... I have most of them.

Thanks!

_______ _____ ____ ____
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_20929.html#21098

> ...
>
> I didn't realize that the BASIS for the Ben Johnston notation was
> alterations from 12-tET... I guess that would make sense from
> what I was shown.

It's not, Joe. The basis of Johnston's notation is the standard
5-limit JI major scale:

A --- E --- B
/ \ / \ / \
/ \ / \ / \
F --- C --- G --- D

All accidentals represent deviations from this.

# and b = +/- 25:24
+ and - = +/- 81:80
7 and L = +/- 36:35

etc.

Johnston's notation makes no reference at all to 12-tET.

Neither do the Pythagorean-based notations of Daniel Wolf and
myself.

> > > OK... so how does a performer know when they have reached
> > > that comma "adjustment?" Is it the absence of beats??
> >
> > Not necessarily! Depends on the chords written. And in Ben
> > Johnston's notation, even the interval D-A with no adjustments
> > beats like a hummingbird's wings.
> >
>
> I see... and that's, of course, 12-tET again...

Sorry, Joe... wrong again.

Paul is talking about the fact that the interval D:A (which is
how I wish Paul had written it, to avoid confusion with Johnston's
minus sign accidental) is not a 3:2, but a 40:27. The 3:2 above
D is A+, and the 3:2 below A is D-. It's all 5-limit JI.

As I stated in my previous post on this, the performer can only
know when he has reached the proper tuning by having a previous
understanding of exactly how the system of accidentals works.
Then, he/she *checks* the intonation by the listening for the
absence of beating, since Johnston almost always uses vertical
sonorities that are Partchian o- or u-tonalities. In other
words, he hardly ever puts intervals like D:A in his music;
a vertical chord would typically have a 3:2 of D-:A or D:A+.

Of course, Paul's arguement becomes valid if we're talking
about Johnston's notation being used by another composer who
*does* use these kinds of intervals (like if I used his notation
for my music, for instance).

>
>
> > > Additionally, how does a COMPOSER know which commas go
> > > where? Are there charts someplace of such? Or is it something
> > > I'm supposed to already know??

There are explanations of Johnston's notation with diagrams in:

- my book (JustMusic)

- David Doty's _Just Intonation Primer_, and _1/1_ articles

- _Perspectives of New Music_ 1996

- Kyle Gann's website

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

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_20929.html#21144

> --- In tuning@y..., "monz" <MONZ@J...> wrote:
>
> /tuning/topicId_20929.html#21054
>
> > Once again, Joe, I refer you to my book (which you have).
> > I devote a whole chapter to Johnston's notation and theory.
> >
>
> Thanks, Monz for the reference!
>
> Yes, I checked your book here... however, the "chapter" you
> mention is only 4 pages in length!

OK... so I should have said "short chapter" instead of
"whole chapter".

> ...
>
> > The notation by Daniel Wolf appeared in a letter in _1/1_.
>
> I'd better hunt around for this... Do you remember the issue
> number.... I have most of them.

Wolf, Daniel. 1996.
Letter in 1/1, the journal of the Just-Intonation Network, 9:3
[Summer], p 15.

This is the first reference listed at the end of my article,
which describes my very similar notation:
http://www.ixpres.com/interval/monzo/article/article.htm

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

--- In tuning@y..., "monz" <MONZ@J...> wrote:
> --- In tuning@y..., jpehrson@r... wrote:
>
> /tuning/topicId_20929.html#21098
>
> > ...
> >
> > I didn't realize that the BASIS for the Ben Johnston notation was
> > alterations from 12-tET... I guess that would make sense from
> > what I was shown.
>
>
> It's not, Joe. The basis of Johnston's notation is the standard
> 5-limit JI major scale:
>
> A --- E --- B
> / \ / \ / \
> / \ / \ / \
> F --- C --- G --- D
>
> All accidentals represent deviations from this.
>
> # and b = +/- 25:24
> + and - = +/- 81:80

To immediately see how the entire 5-limit lattice can be notated with
this method, look at the cover page of my new paper: go to

/tuning/files/perlich/formsoftonality/

and download coverpage.ZIP.

The paper is available from me for $5 (which not quite covers color
copying, binding, and double postage (since I originally sent the
papers to the MicroFest but they arrived too late to be included).

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

/tuning/topicId_20929.html#21151

> There are explanations of Johnston's notation with diagrams in:
>
> - my book (JustMusic)
>
> - David Doty's _Just Intonation Primer_, and _1/1_ articles
>
> - _Perspectives of New Music_ 1996
>
> - Kyle Gann's website

Oops!... Here's the correct citation for the _Perspectives_
article:

Fonville, John. 1991.
"Ben Johnston's Extended Just Intonation: A Guide for Interpreters".
_Perspectives of New Music_, vol 29, no 2 [Summer], p 106-137.

However, there *was* another issue of _Perspectives_ which
was devoted to microtonality, and I believe it was in 1996.
Johnston figures prominently in it.

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

BTW on this subject . I believe Gardner read(?) wrote a book on Microtonal notation i ran across
at a tower records (of all places). Any one seen or know if it is still in print.

monz wrote:

>
> --- In tuning@y..., "monz" <MONZ@J...> wrote:
>
> /tuning/topicId_20929.html#21151
>
> > There are explanations of Johnston's notation with diagrams in:
> >
> > - my book (JustMusic)
> >
> > - David Doty's _Just Intonation Primer_, and _1/1_ articles
> >
> > - _Perspectives of New Music_ 1996
> >
> > - Kyle Gann's website
>
> Oops!... Here's the correct citation for the _Perspectives_
> article:
>
> Fonville, John. 1991.
> "Ben Johnston's Extended Just Intonation: A Guide for Interpreters".
> _Perspectives of New Music_, vol 29, no 2 [Summer], p 106-137.
>
> However, there *was* another issue of _Perspectives_ which
> was devoted to microtonality, and I believe it was in 1996.
> Johnston figures prominently in it.
>
> -monz
> http://www.monz.org
> "All roads lead to n^0"

-- 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:
> BTW on this subject . I believe Gardner read(?) wrote a book on
Microtonal notation i ran across
> at a tower records (of all places). Any one seen or know if it is
still in print.

I saw this book in the library a long time ago and it is my humble
opinion that you wouldn't like it. For example, it essentially treats
Partch's 43-tone notation as if it applied to, of all things, 43-
equal!

In a message dated 4/16/01 5:04:41 PM Eastern Daylight Time,
PERLICH@ACADIAN-ASSET.COM writes:

> I saw this book in the library a long time ago and it is my humble
> opinion that you wouldn't like it. For example, it essentially treats
> Partch's 43-tone notation as if it applied to, of all things, 43-
>

Paul, this is not true. Gardner knows the difference. There has been
controversy with this book and Rudolf Rasch along similar lines. There is
the disappointment that the book does not show the relationship of notation
to function, thereby sidestepping the important issues that are discussed on
this list.

Actually, Gardner really was hunting down every single example of a unique
notation tied to microtonal music that he could. He stayed around the corner
from me for 3 days, and he'd visit with me to look over the archive and he
included everything. (Keep in mind that Gardner climbed my stairs in his
90s).

If you want to see all she wrote from the photograph in time that is
Gardner's book, it will be well worthwhile. The bit about Partch is fully
accurate. There are for Read 43 items to be notated and suggests the
following:

"IThe ratio numbers chosen by Partch express their relationships to his
so-called "generating tone," which is one-line G (392 Hz). From this note
the composer constructed a "resource bank" of pitches (a more accurate term,
one thinks, than "scale") based on the prime numbers 2, 3, 5, 7, 9, and 11,
and their multiples. By combining the resulting twenty-nine pitches obtained
from the 3-limit (3/2, perfect fifth and 4/3, or (major sixth), and 6/5, or
minor third, the 7-, 9-, and 11-limits--all in approximate equal
temperament--and adding fourteen new pitches, also related to the generating
tone of one-line G (1/1) and its octave, 2/1, Partch arrived at his 43-tone
scale or resource bank."

Then Gardner Reed proceeds to give Partch's cents to ratios chart for the
octave from Da Capo Press. I think that Gardner compared Partch's 43 just
with 43-tET and found that 29 pitches were musically interchangeable, while
14 others were "aliens from another planet" new. Keep in mind that Gardner
Read is the internationally recognized notation specialist long before this
book. He's a fabulous composer, lyrical and romantic.

Now to a bassoon student. Johnny Reinhrd

Paul!
I assume you know that it is not the "equal" part that i find disturbing, but the 43. 41 with
a little added sign for the variations at two places would be fine with me. maybe an ideal as an
alternate to the tableture. I used it when studying Castor and Pollux.
I do remember Read mentioning Erv's method of notation based on how a scale maps to a
keyboard. Considering it was the first type of notation I have used and exposed I have not felt a
reason to change to another.

PERLICH@ACADIAN-ASSET.COM wrote:

> --- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> > BTW on this subject . I believe Gardner read(?) wrote a book on
> Microtonal notation i ran across
> > at a tower records (of all places). Any one seen or know if it is
> still in print.
>
> I saw this book in the library a long time ago and it is my humble
> opinion that you wouldn't like it. For example, it essentially treats
> Partch's 43-tone notation as if it applied to, of all things, 43-
> equal!

-- 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:
> Paul!
> I assume you know that it is not the "equal" part that i find
disturbing, but the 43. 41 with
> a little added sign for the variations at two places would be fine
with me.

As you probably know, we (you and I) are of one mind on this, Kraig!

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

/tuning/topicId_20929.html#21054

I've been studying through both Monz' materials and the Daniel Wolf
letter in 1/1 regarding "improvements" in the Ben Johnston notation...

Well, it really seems as though Wolf has a brilliant suggestion here.
The implications from his letter is that the Ben Johnston notation
doesn't really "work." Is that correct??

He says, depending on the fundamental, some of the supposedly
"just" intervals... D to A, in particular, would vary... in some cases
even being 40/27!

Using a Pythagorean chain as a basis, since it is the foundation of
the staff system, really makes a lot of sense, and it is totally
transposible.... Wow.

But, composers such as David Doty and Kyle Gann are writing big pieces
in the Ben Johnston system.

Is this to say that their pieces really aren't coming out as they
think they are???

That would be a little discouraging, to say the least!

____________ _______ ___ _
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

Hi Joseph,

> He says, depending on the fundamental, some of the supposedly
> "just" intervals... D to A, in particular, would vary... in some
cases
> even being 40/27!

I'm pretty sure D to A would _always_ be 40/27 . . . is that not
correct?
>
> Using a Pythagorean chain as a basis, since it is the foundation of
> the staff system, really makes a lot of sense, and it is totally
> transposible.... Wow.

Yup!!!
>
> But, composers such as David Doty and Kyle Gann are writing big
pieces
> in the Ben Johnston system.

Well, it depends who's supposed to read them.
>
> Is this to say that their pieces really aren't coming out as they
> think they are???

I'm sure David Doty and Kyle Gann would be able to hear that if it
were the case about their pieces. These guys don't just write ratios
without knowing _exactly_ what they're supposed to sound like. A look
into their writings should explain how they could know this . . .
they're probably the best two guys to explain it around!

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

/tuning/topicId_20929.html#21151

>
> It's not, Joe. The basis of Johnston's notation is the standard
> 5-limit JI major scale:
>
> A --- E --- B
> / \ / \ / \
> / \ / \ / \
> F --- C --- G --- D
>
> All accidentals represent deviations from this.
>
> # and b = +/- 25:24
> + and - = +/- 81:80
> 7 and L = +/- 36:35
>
> etc.
>

Thank you so much, Monz for helping to explain this. Right off the
top, I would like to say that I am casting no aspersions on the fine
compositional work of BOTH David Doty and Kyle Gann. In fact, I own
the Doty CD "Uncommon Practices."

The problem is this: spread out in front of me I have YOUR book, the
1/1 article of Daniel Wolf and two examples of compositions that I got
from the Microfest, by Kyle Gann and David Doty. These are BOTH in
Ben Johnston's notation.

At the moment, I am rather fascinated by the IMPLICATIONS of the
Johnston notation... the idea that somehow maybe a player could adjust
his performance by one of the various commas indicated in the score.

This would, of course, be a much more AURAL process than writing some
kind of cents notation. The thought of this has me mesmerized at the
moment.

HOWEVER, I am not fully convinced that a performer could do this.
Maybe if the tempo were VERY slow, but how about in fast passages?

Does anybody know the success of Ben Johnston in making this happen
with real, live instrumentalists?? Perhaps I need to know more about
Johnston.

I met Johnston once, and he was actually not all that friendly. But
neither was John Harbison... so much for that.

Anyway, I OBVIOUSLY need more help with trying this out, and I am
encountering many problems. David Doty says I need a book that nobody
has yet written... I've tried to get the "gist" of composing in his
system from his scores that I have here but, so far, to no avail.

Kyle Gann's "harmonic skeletons" are MORE helpful. At least I can get
some kind of glimmer about what is going on and I know what the
fundamental is... but I still would have problems putting it all
together, if I wanted to write using this notation.

And THEN, there are the cloying problems cited by Daniel Wolf in his
article in 1/1. To quote:

"The 'hybrid' notations (for example, Ben Johnston's) substitute
ratios with the factor five for the pure Pythagorean sequence and
consequently lose the property of interval invariance. Thus, without
ancillary notation of the local "tonic," the exact ratio of an
interval is uncertain. [ED.-- DIS NO PLEASE ME MUCHO] (In some
contexts, an unmodified fifth D-A may have the ratio 3:2, in others
40:27)..."

This makes a "doubting Thomas" like me, well, doubt...

I'm wondering if there is any book that could help me? Does anybody
know about the book available through the JI Network called _The Music
of Ben Johnston_ by Heidi Von Gunden? Does this have much about his
notational system and PRACTICAL ways of composing with it?? At least
maybe it tells about performances... (??) I am STILL unconvinced that
performers could really use this system even if a composer could
figure it out. For me, I would need some kind of charts or more
systematic instruction in order to work practically with it.

It's not like I'm "well, isn't this theory interesting... now on to
the next thing..." I would really want to understand it in order to
COMPOSE with it, not just regard it as a curiosity...

The fact that Kyle Gann and David Doty BOTH use it extensively augurs
well for it as a system. HOWEVER, I am under the impression that BOTH
of them use it MOSTLY with ELECTRONIC works! I could be wrong, but
that is the impression I have. I have NO idea how much practical
experience they have had with "real" performers with it...

Now on to at least a couple of specifics:

Monz, in your book you show the various alterations for the commas
from the 5, 7 and 11 limit.

The 5-limit syntonic comma is obvious: 81:80 with the + and -

The 7-limit is clear with his use of 7's and inverted 7's: 36:35

AND, the 11-limit is the 33:32 with the arrows.

SO, what's the 25:24 all about??

See, I don't even know THIS much. How am I going to compose with
this...

I'm going to keep studying over this stuff... but I really need some
kind of systematic book or chart or some such, to get going with this.

AND, I'm not even certain that players can play it or that it really
"works" with the Wolf caveats.

This is a problem, man, a problem...

>
> Sorry, Joe... wrong again.
>
> Paul is talking about the fact that the interval D:A (which is
> how I wish Paul had written it, to avoid confusion with Johnston's
> minus sign accidental) is not a 3:2, but a 40:27. The 3:2 above
> D is A+, and the 3:2 below A is D-. It's all 5-limit JI.
>
> As I stated in my previous post on this, the performer can only
> know when he has reached the proper tuning by having a previous
> understanding of exactly how the system of accidentals works.

Well, first the COMPOSER has to know it, and then he has to TEACH it
to the performer.

Good luck to all concerned....

> Then, he/she *checks* the intonation by the listening for the
> absence of beating, since Johnston almost always uses vertical
> sonorities that are Partchian o- or u-tonalities.

Well, that's the fascinating part, if it really works.

In other
> words, he hardly ever puts intervals like D:A in his music;
> a vertical chord would typically have a 3:2 of D-:A or D:A+.
>
> Of course, Paul's arguement becomes valid if we're talking
> about Johnston's notation being used by another composer who
> *does* use these kinds of intervals (like if I used his notation
> for my music, for instance).
>
> >
> >
> > > > Additionally, how does a COMPOSER know which commas go
> > > > where? Are there charts someplace of such? Or is it something
> > > > I'm supposed to already know??
>
>
> There are explanations of Johnston's notation with diagrams in:
>
> - my book (JustMusic)

Monz.. this is only 4 pages!

>
> - David Doty's _Just Intonation Primer_, and _1/1_ articles

There is not enough in the JI Primer to help the composer with this...
Doty admits it himself. It just describes the system, as YOU do.

One would need a book with systematic charts and examples, or maybe
personal study with Johnston, as Kyle Gann undertook....

>
> - _Perspectives of New Music_ 1996

OK, I haven't seen this one...

>
> - Kyle Gann's website

I'm not finding this on his site, and I don't remember it being on his
site. Do you have the direct link.
>
>

Don't worry... I'm doing great with this. Now I can just go and
commit suicide...

(Just joking...)

___________ ______ ____ _
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_20929.html#21194

> --- In tuning@y..., "monz" <MONZ@J...> wrote:
>
> /tuning/topicId_20929.html#21054
>
> [monz post snipped]
>
> I've been studying through both Monz' materials and the Daniel
> Wolf letter in 1/1 regarding "improvements" in the Ben Johnston
> notation...
>
> Well, it really seems as though Wolf has a brilliant suggestion
> here.

I have to agree - since Wolf's is essentially the same as mine. :)

> The implications from his letter is that the Ben Johnston
> notation doesn't really "work." Is that correct??
>
> He says, depending on the fundamental, some of the supposedly
> "just" intervals... D to A, in particular, would vary... in some
> cases even being 40/27!

That's a bit muddled, Joe. What Wolf and I both say is that
a notated interval that a performer would automatically *assume*
to be just is not always so in Johnston's notation.

> Using a Pythagorean chain as a basis, since it is the foundation of
> the staff system, really makes a lot of sense, and it is totally
> transposible.... Wow.

It makes more sense because - in addition to the two points
you make - it's more logical, more consistent, simpler, more
intuitive, etc. etc.

> But, composers such as David Doty and Kyle Gann are writing
> big pieces in the Ben Johnston system.

And I think that's too bad - they're missing the opportunity
to put the better Wolf/Monzo notation to work.

(BTW, this would be a good place to note that Wolf and I are
certainly not the only two who have advocated this approach.
It's also been used by Clarence Barlow and Doug Keislar in
their writings - I don't know about their scores...)

> Is this to say that their pieces really aren't coming out as they
> think they are???
>
> That would be a little discouraging, to say the least!

No, Joe. As should have been apparent from the Doty and Gann
lectures at Microfest, those two certainly know Johnston's
notation inside and out, and are well aware of how to notate
every interval they want precisely.

The problem is simply that Johnston's notation is harder to
grasp than the alternative Pythagorean-based one.

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

Apologies in advance for the length of this.

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_20929.html#21237

> --- In tuning@y..., "monz" <MONZ@J...> wrote:
>
> /tuning/topicId_20929.html#21151
>
> >
> > It's not, Joe. The basis of Johnston's notation is the standard
> > 5-limit JI major scale:
> >
> > A --- E --- B
> > / \ / \ / \
> > / \ / \ / \
> > F --- C --- G --- D
> >
> > All accidentals represent deviations from this.
> >
> > # and b = +/- 25:24
> > + and - = +/- 81:80
> > 7 and L = +/- 36:35
> >
> > etc.
> >
>
> Thank you so much, Monz for helping to explain this.
>
> ...
>
> At the moment, I am rather fascinated by the IMPLICATIONS of the
> Johnston notation... the idea that somehow maybe a player could
> adjust his performance by one of the various commas indicated
> in the score.
>
> This would, of course, be a much more AURAL process than
> writing some kind of cents notation. The thought of this has
> me mesmerized at the moment.
>
> HOWEVER, I am not fully convinced that a performer could do this.
> Maybe if the tempo were VERY slow, but how about in fast passages?
>
> Does anybody know the success of Ben Johnston in making this
> happen with real, live instrumentalists?? Perhaps I need to
> know more about Johnston.

As I tried to emphasize in my previous post, Joe, Johnston has
fairly good success with his method *because* the music he
writes is permeated with vertical JI sonorities (o- and
u-tonalities).

Therefore, once the performers understand both the notation
and the compositional style, it's pretty easy to get decent
results. Of course, results vary widely depending on the
performer...

I met Johnston in New York in (IIRC) December 1996, when Eric
Grunen and a small orchestra performed his _Chamber Symphony_.
I attended the rehearsal, and, while I was unfamiliar with the
score previously, as I looked at it, it seemed to me that
most of the performers were navigating the intonational field
quite admirably.

The caveat in all this: Johnston's music assumes that 5-limit
scale quoted above as its basis. If your music doesn't do this
(for example, mine frequently does not), then Johnston's notation
will probably not work too well for your music.

> I'm wondering if there is any book that could help me? Does
> anybody know about the book available through the JI Network
> called _The Music of Ben Johnston_ by Heidi Von Gunden? Does
> this have much about his notational system and PRACTICAL ways
> of composing with it?? At least maybe it tells about
> performances... (??)

Heidi's book is mostly a detailed overview of Johnston's life
and music. Since it does cover his compositions quite
comprehensively, it can't help but have a lot to say about
his notation.

But the best source for a deep understanding of Johnston's
notation is the Fonville _Perspectives_ article. (full citation
given below)

If you're going to dig, it would also be useful to read
Johnston's own initial articles on all this:

Johnston, Ben. 1964.
"Scalar Order as a Compositional Resource".
_Perspectives of New Music_, vol 2 no 2 [Spring-Summer], p 56-76.

Johnston, Ben. 1966.
"Proportionality and Expanded Musical Pitch Relations".
_Perspectives of New Music_, vol 5 no 1 [Fall-Winter], p 112-120.

You can get some philosphical background there that you won't
find elsewhere.

> I am STILL unconvinced that performers
> could really use this system even if a composer could figure
> it out. For me, I would need some kind of charts or more
> systematic instruction in order to work practically with it.
>
> It's not like I'm "well, isn't this theory interesting... now on to
> the next thing..." I would really want to understand it in order to
> COMPOSE with it, not just regard it as a curiosity...
>
> The fact that Kyle Gann and David Doty BOTH use it extensively
> augurs well for it as a system. HOWEVER, I am under the
> impression that BOTH of them use it MOSTLY with ELECTRONIC
> works! I could be wrong, but that is the impression I have.
> I have NO idea how much practical experience they have had
> with "real" performers with it...

Please keep in mind the caveat I wrote above. Johnston's notation
works very well for both Gann and Doty *because* they compose
in JI, and that 5-limit basic scale really *is* a basic part of
their musical vocabulary.

Based on what I know of your own compositions, Joe, it seems to
me that the Wolf/Monzo Pythagorean-based notation would work
better for you. I've never known any of your music to prominently
feature 5-limit JI. Based on the logical superiority of the
Pythagorean-based approach, I think you would favor it.

>
> Now on to at least a couple of specifics:
>
> Monz, in your book you show the various alterations for the commas
> from the 5, 7 and 11 limit.
>
> The 5-limit syntonic comma is obvious: 81:80 with the + and -
>
> The 7-limit is clear with his use of 7's and inverted 7's: 36:35
>
> AND, the 11-limit is the 33:32 with the arrows.
>
> SO, what's the 25:24 all about??
>
> See, I don't even know THIS much. How am I going to compose with
> this...

I think it's because somehow you keep trying to relate it all
to 12-tET, which won't work here.

Here's a quick tutorial:

Johnston's notation assumes that 7-tone 5-limit scale as its
basis, which means that the 7 letter names with no other
accidentals represent that scale: A B C D E F G.

The sharps and flats in Johnston's system represent an alteration
of any of those 7 notes by a 25:24. Thus, if C = 1/1, then
C# = 25/24. If E = 5/4, then Eb = 6/5. Etc.

The big problem I have is that, after 5, all of Johnston's
accidentals logically and consistently represent a *prime-factor*
intonational inflection, but he muddled this distinction when
dealing with prime-factors 3 and 5. So the plusses and minuses
(+/-) represent the 81:80 syntonic comma, but in a "weird"
pattern than can really only be understood by invoking lattice
visualizations.

It would be better to have the plain letters A B C D E F G
to represent *only* prime-factor 3, as it already does
historically in our standard notation.

Then, the sharps and flats would represent an inflection of
2187:2048 (== 3^7), the usual Pythagorean "chromatic semitone"
or "apotome", again, as it already does historically in our
standard notation.

The two advantages to this:

1)
Elimination of the presence of prime-factor 5 in the sharps
and flats, so that they only indicate intonational adjustments
involving prime-factor 3, and so they too are on the same footing
as all other accidentals.

2)
The presence of a plus or minus *always* indicates the presence
of prime-factor 5. This puts the +/- accidentals on the same
footing as all the others (for 7, 11, 13, etc.).

In Johnston's notation, both the #/b *and* +/- accidentals
*always* indicate an adjustment which includes *both*
prime-factors 3 and 5, because his basic scale includes both.
This is my primary difficulty with his system.

Johnston's approach is fairly easy to conceptualize IF THE
RESOURCES REMAIN FAIRLY SMALL, which is not the case even
in his own music.

The Pythagorean-based approach is even simpler to conceptualize,
but *MUCH* simpler to extend, because of its linearity, and
because of the rigid consistency of its "rules" in regard to
the meaning of accidentals.

> > [me, monz:]
> >
> > Then, he/she *checks* the intonation by the listening for the
> > absence of beating, since Johnston almost always uses vertical
> > sonorities that are Partchian o- or u-tonalities.
>
>
> Well, that's the fascinating part, if it really works.
>
>
> In other
> > words, he hardly ever puts intervals like D:A in his music;
> > a vertical chord would typically have a 3:2 of D-:A or D:A+.

The notation is adapted to the kind of music written by the
composer who invented it. Same as with Partch. It works
for Johnston, and for several other JI composers, but that
doesn't automatically mean that it would be well-suited for
you or anyone else.

> > - _Perspectives of New Music_ 1996
>
>
> OK, I haven't seen this one...

That's the one you need. It's quite comprehensive.
I had the date wrong - please note the correct citation:

Fonville, John. 1991.
"Ben Johnston's Extended Just Intonation: A Guide for Interpreters."
_Perspectives of New Music_, vol 29, no 2 [Summer], p 106-137.

>
> >
> > - Kyle Gann's website
>
>
> I'm not finding this on his site, and I don't remember it being
> on his site. Do you have the direct link.

Sorry, Joe... I thought I remembered Kyle having a discussion
of it on his site. I haven't checked recently... I could be wrong.

Of course, the best way to understand all this is to vault over
the theoretical articles and study the scores themselves.
Johnston has *many* scores published, by Smith Publications.
I know for a fact that the Philadelphia Free Library has several
of them, which you should be able to get thru interlibrary loan.

The problem is that a lot of his work has not been recorded,
and that which has, as you might imagine, has somewhat
questionable intonation.

Two recommendations:

1)
The most easily available Johnston recording is his _4th Quartet_
(variations on "Amazing Grace" - it goes up to 7-limit) on the
Kronos Quartet CD _White Man Sleeps_. Big chunks of this
score also appears in the following article:

Shinn, Randall. 1977.
"Ben Johnston's Fourth String Quartet".
_Perspectives of New Music_, vol 15 no 2 [Spring-Summer], p 145-173).

Note that this piece is especially a good starting point for you
because I give lattices of its scales in my book.

2)
In my opinion, the most outstanding example of Johnston's ability
and talent is his _6th Quartet_. A great analysis of this was
given in:

Elster, Steven. 1991.
"A Harmonic and Serial Analysis of Ben Johnston's
String Quartet No. 6".
_Perspectives of New Music_, vol 29 no 2 [Summer], p 138-165.

(Note that this is the same issue as the Fonville article.)

There was a recording of this on vinyl, unfortunately now long
out of print. Again, I believe I tracked down a copy of it
at the Philadelphia Library. It's a good performance (better,
IMO, than the way Kronos handles Johnston's work).

There, that should keep you busy studying for quite some time...

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

--- In tuning@y..., jpehrson@r... wrote:
>
> SO, what's the 25:24 all about??
>
Hi Joseph.

You may recall, from our off-list conversations a couple of weeks
ago, me explaining to you how the 25:24 is the JI chromatic semitone,
hence Ben Johnston's chromatic symbols (#, b) indicate raising or
lowering by a 25:24. My new paper will clarify this further . . . I
received your $5 check (thanks!) and I will send you a copy as soon
as I get them back from Bill Alves . . .
>
> AND, I'm not even certain that players can play it or that it really
> "works" with the Wolf caveats.
>
>
> This is a problem, man, a problem...

I don't see what the problem is. Who's forcing you to use Johnston's
notation rather than Wolf's notation for your own music? Or you might
eeven want to use 72-tET notation (if you're not moving across vast
stretches of the lattice) or cents notation . . . perfectly good
notations when the final result desired is 11-limit JI harmony . . .
and perhaps easier to train 12-tET-trained musicians with.

And again, if you do want to use Johnston's notation, the cover of my
paper (for those who haven't ordered it yet, the cover appears at
/tuning/files/perlich/formsoftonality/ --
download coverpage.ZIP) shows how Johnston's notation applies within
the 5-limit . . . you'll see three columns of identically notated
notes . . . just add a "+" every time you move one column (an 81:80)
to the right, and a "-" every time you move one column to the left
(+'s and -'s cancel one another out, of course).

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

> And I think that's too bad - they're missing the opportunity
> to put the better Wolf/Monzo notation to work.
>
> (BTW, this would be a good place to note that Wolf and I are
> certainly not the only two who have advocated this approach.
> It's also been used by Clarence Barlow and Doug Keislar in
> their writings - I don't know about their scores...)

It's also essentially identical to the notation that Helmholtz,
Ellis, and just about every JI theorist ever to use letter-names,
used! The only "renegades" seem to have been Johnston and Fokker
(Fokker's D-A is just but doesn't assume that B-F# or Bb-F is
necessarily a just fifth).

--- In tuning@y..., "monz" <MONZ@J...> wrote:
>
> The big problem I have is that, after 5, all of Johnston's
> accidentals logically and consistently represent a *prime-factor*
> intonational inflection, but he muddled this distinction when
> dealing with prime-factors 3 and 5. So the plusses and minuses
> (+/-) represent the 81:80 syntonic comma, but in a "weird"
> pattern than can really only be understood by invoking lattice
> visualizations.

Specifically, the lattice visualization on the cover of my new paper
(see my last post for a link to a free look at the cover).

I agree with everything Monz wrote . . . my new paper may make a
better case for why someone perhaps _would_ want to use Johnston's
notation, though . . . but I agree with Monz that the
Helmholtz/Ellis/Wolf/Monzo notation is much more transparent and
logical.

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

/tuning/topicId_20929.html#21151

> --- In tuning@y..., jpehrson@r... wrote:
>
> /tuning/topicId_20929.html#21098
>

>
> It's not, Joe. The basis of Johnston's notation is the standard
> 5-limit JI major scale:
>
> A --- E --- B
> / \ / \ / \
> / \ / \ / \
> F --- C --- G --- D
>
> All accidentals represent deviations from this.
>
> # and b = +/- 25:24
> + and - = +/- 81:80
> 7 and L = +/- 36:35
>
> etc.
>

Oh, I see, Monz! You mean that the 25:24 is his use of "regular"
accidentals... I wish you had included the simple ratios above and
the diagram in your short chapter about Johnston...

Sorry, I don't mean to be complaining...

> Paul is talking about the fact that the interval D:A (which is
> how I wish Paul had written it, to avoid confusion with Johnston's
> minus sign accidental) is not a 3:2, but a 40:27. The 3:2 above
> D is A+, and the 3:2 below A is D-. It's all 5-limit JI.
>
> As I stated in my previous post on this, the performer can only
> know when he has reached the proper tuning by having a previous
> understanding of exactly how the system of accidentals works.

This is going to be good... since *I* am having enough trouble
figuring it out, how is the reluctant PERFORMER going to do it?!

I'm beginning to think this notation is strictly a "cult item" for the
fully initiated.

Are you sure this isn't really the "Skull and Bones Society" branch of
microtonal aficionados??

________ _____ ____ ___
Joseph Pehrson

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21153

>
> To immediately see how the entire 5-limit lattice can be notated
with
> this method, look at the cover page of my new paper: go to
>
> /tuning/files/perlich/formsoftonality/
>
> and download coverpage.ZIP.
>

This is so very, very, cool....

If ANYTHING is going to help me understand where these commas go,
graphics like this will! Beautiful patterns, too.

And the title, "The Forms of Tonality" is INCREDIBLY inspired! (Simple
and beautiful)

I can't wait to get my copy!!

__________ ______ _____ __
Joseph Pehrson

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

/tuning/topicId_20929.html#21154

>
> --- In tuning@y..., "monz" <MONZ@J...> wrote:
>
> /tuning/topicId_20929.html#21151
>
> > There are explanations of Johnston's notation with diagrams in:
> >
> > - my book (JustMusic)
> >
> > - David Doty's _Just Intonation Primer_, and _1/1_ articles
> >
> > - _Perspectives of New Music_ 1996
> >
> > - Kyle Gann's website
>
>
>
> Oops!... Here's the correct citation for the _Perspectives_
> article:
>
> Fonville, John. 1991.
> "Ben Johnston's Extended Just Intonation: A Guide for Interpreters".
> _Perspectives of New Music_, vol 29, no 2 [Summer], p 106-137.
>
>
> However, there *was* another issue of _Perspectives_ which
> was devoted to microtonality, and I believe it was in 1996.
> Johnston figures prominently in it.
>
>

Hi Monz!

It looks as thought this "Perspectives" article is EXACTLY what I
need... I'm going to try to hunt it up...

_______ _____ _____ _
Joseph Pehrson

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

/tuning/topicId_20929.html#21156

> BTW on this subject . I believe Gardner read(?) wrote a book on
Microtonal notation i ran across
> at a tower records (of all places). Any one seen or know if it is
still in print.
>

Kraig!

Suddenly I am having trouble reading your messages in Netscape 6. It
is only the messages that are very close to the top of the e-mail...

Do you have any idea why that is?? I have to "view the source code"
just to see what's there...

Anybody else know what could be happening??

________ _____ _____
Joseph Pehrson

J>P> ! no idea, i still use 4.6

jpehrson@rcn.com wrote:

> --- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
>
> /tuning/topicId_20929.html#21156
>
> > BTW on this subject . I believe Gardner read(?) wrote a book on
> Microtonal notation i ran across
> > at a tower records (of all places). Any one seen or know if it is
> still in print.
> >
>
> Kraig!
>
> Suddenly I am having trouble reading your messages in Netscape 6. It
> is only the messages that are very close to the top of the e-mail...
>
> Do you have any idea why that is?? I have to "view the source code"
> just to see what's there...
>
> Anybody else know what could be happening??
>
> ________ _____ _____
> Joseph Pehrson

-- 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..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21255

> --- In tuning@y..., "monz" <MONZ@J...> wrote:
>
> > And I think that's too bad - they're missing the opportunity
> > to put the better Wolf/Monzo notation to work.
> >
> > (BTW, this would be a good place to note that Wolf and I are
> > certainly not the only two who have advocated this approach.
> > It's also been used by Clarence Barlow and Doug Keislar in
> > their writings - I don't know about their scores...)
>
> It's also essentially identical to the notation that Helmholtz,
> Ellis, and just about every JI theorist ever to use letter-names,
> used! The only "renegades" seem to have been Johnston and Fokker
> (Fokker's D-A is just but doesn't assume that B-F# or Bb-F is
> necessarily a just fifth).

True.

I'm pretty sure that the weirdness of Johnston's notation is
a remnant of its development during the early stages of his
work in JI. For quite a while in the 1960s, he stuck to a
very extended (IIRC, 53-tone) 5-limit JI. It was only during
the 1970s that he expanded to 7-limit, and the 1980s and 1990s
that he went thru 13-limit and eventually all the way to 31.

Coupled with Johnston's stated desire to start with a basic
scale which he felt would be easy to grasp intuitively by
any trained musician, that's the only explanation I can give,
because anyone who starts using lattices to represent ratios
will come up with the Pythagorean-based approach as the best way.

Considering this thought, I have no idea why Fokker's notation
is similarly weird, since he was the first modern "lattice master".

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

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21257

> --- In tuning@y..., "monz" <MONZ@J...> wrote:
> >
> > The big problem I have is that, after 5, all of Johnston's
> > accidentals logically and consistently represent a *prime-factor*
> > intonational inflection, but he muddled this distinction when
> > dealing with prime-factors 3 and 5. So the plusses and minuses
> > (+/-) represent the 81:80 syntonic comma, but in a "weird"
> > pattern than can really only be understood by invoking lattice
> > visualizations.
>
> Specifically, the lattice visualization on the cover of my new
> paper (see my last post for a link to a free look at the cover).
>
> I agree with everything Monz wrote . . . my new paper may make a
> better case for why someone perhaps _would_ want to use Johnston's
> notation, though . . . but I agree with Monz that the
> Helmholtz/Ellis/Wolf/Monzo notation is much more transparent and
> logical.

WAYYYYYYYYYY COOL !!!

It's such a good feeling to have you agree wholeheartedly with
me for a change, with zero debate !!!!! (Is this a first?)

-monz

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_20929.html#21260

> --- In tuning@y..., "monz" <MONZ@J...> wrote:
>
> /tuning/topicId_20929.html#21151
>
> > --- In tuning@y..., jpehrson@r... wrote:
> >
> > /tuning/topicId_20929.html#21098
> >
>
> >
> > It's not, Joe. The basis of Johnston's notation is the standard
> > 5-limit JI major scale:
> >
> > A --- E --- B
> > / \ / \ / \
> > / \ / \ / \
> > F --- C --- G --- D
> >
> > All accidentals represent deviations from this.
> >
> > # and b = +/- 25:24
> > + and - = +/- 81:80
> > 7 and L = +/- 36:35
> >
> > etc.
> >
>
> Oh, I see, Monz! You mean that the 25:24 is his use of "regular"
> accidentals... I wish you had included the simple ratios above and
> the diagram in your short chapter about Johnston...
>
> Sorry, I don't mean to be complaining...

It's OK, Joe... Those who own my book are already extremely
well aware of what I've been reiterating for several years now:
IT'S STILL NOT FINISHED!

There will eventually be at least a little more on Johnston,
including my own examination (with lattices) of his _6th Quartet_.

> > Paul is talking about the fact that the interval D:A (which
> > is how I wish Paul had written it, to avoid confusion with
> > Johnston's minus sign accidental) is not a 3:2, but a 40:27.
> > The 3:2 above D is A+, and the 3:2 below A is D-. It's all
> > 5-limit JI.
> >
> > As I stated in my previous post on this, the performer can only
> > know when he has reached the proper tuning by having a previous
> > understanding of exactly how the system of accidentals works.
>
>
> This is going to be good... since *I* am having enough trouble
> figuring it out, how is the reluctant PERFORMER going to do it?!
>
> I'm beginning to think this notation is strictly a "cult item"
> for the fully initiated.

That's largely how I feel about it.

> Are you sure this isn't really the "Skull and Bones Society"
> branch of microtonal aficionados??

Well, as I've said, this notation works for Johnston, and he's
been happy with it for several decades now. And if others find
it useful, good for them.

I just think it's unfortunate that it seems to be slowly becoming
the default JI staff-based notation, when there is a clearly
superior alternative.

As I've said, Wolf's notation is essential identical to mine,
at least conceptually. I know now that you like it, Joe, but
I really don't see the need for the strange typographical symbols,
even tho I admit that they have a nice compactness which
facilitates reading the notation.

I know for sure that I'll never give up my plain ol' number-based
prime-factor notation. For good examples, take a look at my
transcription of Partch's _The Intruder_ (in my book), or of
my analysis of Robert Johnson's vocal line to _Drunken Hearted Man_
(supplemented with Reinhard-style cents deviations from 12-tET), at:
http://www.ixpres.com/interval/monzo/rjohnson/drunken.htm

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

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_20929.html#21263

> --- In tuning@y..., "monz" <MONZ@J...> wrote:
>
> /tuning/topicId_20929.html#21154
>
> >
> > Fonville, John. 1991.
> > "Ben Johnston's Extended Just Intonation: A Guide for
> > Interpreters".
> > _Perspectives of New Music_, vol 29, no 2 [Summer], p 106-137.
>
>
> Hi Monz!
>
> It looks as thought this "Perspectives" article is EXACTLY what I
> need... I'm going to try to hunt it up...

You shouldn't have a problem finding it in New York. But if you
do, and if your interlibrary loan works with a private college
library, you can find it at Connelly Library, of LaSalle University,
in Philadelphia.

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

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

> because anyone who starts using lattices to represent ratios
> will come up with the Pythagorean-based approach as the best way.
>
> Considering this thought, I have no idea why Fokker's notation
> is similarly weird, since he was the first modern "lattice master".

Well, Fokker was a theoretician first and, while his 31-tone notation
is of course eminently practical for performance, in his JI
explorations the practicality of the notation for performance never
became a consideration -- rather, the notation was chosen so as to
minimize the quantity of extra symbols needed in his theoretical
writings. First, Fokker started with a 7-tone Pythagorean scale from
F to B. Then, Fokker's rule was this: D is always at the center of
the lattice, and if two notes in the lattice are a comma (of whatever
variety) apart, the one closer to the center is notated without any
special symbol. Hence, F# refers to the major third above D, while
the fifth above B Fokker refers to as /F#.

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

> > I agree with everything Monz wrote . . . my new paper may make a
> > better case for why someone perhaps _would_ want to use
Johnston's
> > notation, though . . . but I agree with Monz that the
> > Helmholtz/Ellis/Wolf/Monzo notation is much more transparent and
> > logical.
>
>
>
> WAYYYYYYYYYY COOL !!!
>
> It's such a good feeling to have you agree wholeheartedly with
> me for a change, with zero debate !!!!! (Is this a first?)
>

We've agreed on this very issue umpteen times in the past . . . did
you forget?

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21288

> --- In tuning@y..., "monz" <MONZ@J...> wrote:
>
> > because anyone who starts using lattices to represent ratios
> > will come up with the Pythagorean-based approach as the best way.
> >
> > Considering this thought, I have no idea why Fokker's notation
> > is similarly weird, since he was the first modern "lattice
> > master".
>
> Well, Fokker was a theoretician first and, while his 31-tone
> notation is of course eminently practical for performance, in
> his JI explorations the practicality of the notation for
> performance never became a consideration -- rather, the notation
> was chosen so as to minimize the quantity of extra symbols needed
> in his theoretical writings. First, Fokker started with a 7-tone
> Pythagorean scale from F to B. Then, Fokker's rule was this: D
> is always at the center of the lattice, and if two notes in the
> lattice are a comma (of whatever variety) apart, the one closer
> to the center is notated without any special symbol. Hence, F#
> refers to the major third above D, while the fifth above B Fokker
> refers to as /F#.

Thanks, Paul. That seems like a plausible explanation to me.

(Boy, we sure are seeing eye-to-eye a lot these days!
What's going on?...)

-monz

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

/tuning/topicId_20929.html#21246
>

> > [monz post snipped]
> >
> > I've been studying through both Monz' materials and the Daniel
> > Wolf letter in 1/1 regarding "improvements" in the Ben Johnston
> > notation...
> >
> > Well, it really seems as though Wolf has a brilliant suggestion
> > here.
>
>
> I have to agree - since Wolf's is essentially the same as mine. :)
>

It seems that Wolf's notation gets a little bizarre in terms of
accidental usage, now?? It looks a little as though somebody fell
the wrong way on the typewriter....

AND, the "Monzotone" notation is still beyond me. I have read that
page 20 times and I still don't understand it.

Is it possible, Monz, to describe your Just system in some kind of
terms so that somebody like me can understand it?? It seems a little
obfuscatory...

I will try to find the Perspectives article on Ben Johnston's
notation... although I am beginning to doubt that it is as "direct"
as these "improved" systems.

The reason, though, that I am so interested in it is, obviously,
Johnston's stature as a composer and xenharmonic thinker and the
usage of such composers as Doty and Gann...

It can't just be "overlooked..."

________ _____ ____ _
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

> It seems that Wolf's notation gets a little bizarre in terms of
> accidental usage, now?? It looks a little as though somebody fell
> the wrong way on the typewriter....

Isn't that equally the case with Johnston's notation?
> >
> I will try to find the Perspectives article on Ben Johnston's
> notation... although I am beginning to doubt that it is as "direct"
> as these "improved" systems.
>
> The reason, though, that I am so interested in it is, obviously,
> Johnston's stature as a composer and xenharmonic thinker and the
> usage of such composers as Doty and Gann...
>
I don't think either of these men would compose any differently if
they happened to use Helmholtz/Ellis/Wolf/Monzo notation rather than
Johnston's. They both have a clear grasp (and have provided the two
clearest explanations) of what JI is all about. JI is about sound (at
least to many of us -- i know Dante Rosati has a different
philosophy, so we should remain respectful of that). Perhaps some
serial composition is more about notes on a page than about sound.
But to these two men, JI is about sound. These men would simply write
in ratios if that were a useful notation. It wouldn't affect the
music they wrote.

By the way, did you catch David Doty's reply to you on this list?

But you still haven't said much about how you expect yout notation to
be used. Who's going to be reading it? How can you best make use of
the training they already have? How can you minimize the number of
new concepts they have to learn? Those are the questions i would be
asking, rather than worrying about anyone's "stature".

Anyway, here are some just major and minor triads in Johnston's
notation:

Bb- D- F (major)

Bb D F+ (major)

B- D- F# (minor)

B D F#+ (minor)

C E G (major)

C Eb G (minor)

D- F# A (major)

D F#+ A+ (major)

D- F A (minor)

D F+ A+ (minor)

Now the same triads in Wolf's notation:

Bb D- F (major)

Bb+ D F+ (major)

B- D F#- (minor)

B D+ F# (minor)

C E- G (major)

C Eb+ G (minor)

D F#- A (major)

D+ F# A+ (major)

D- F A- (minor)

D F+ A (minor)

Note that in Wolf's notation, the root and fifth always have the same
commatic accidental applied to them; the minor third has an extra +
(or one less -), and the major third has an extra - (or one less +).
These conventions are easy to remember since the alterations are
measured from the Pythagorean basis of Wolf's notation.

In Johnston's notation, there is no consistent pattern to the way the
triads are notated. You have to look at a lattice diagram to know
where you are. Even if you take for granted that all triads in a
score are going to sound just, reckoning the melodic intervals from
one to the other is still going to require you to stop and look at
the lattice (until you have the process memorized).

Of course, Joseph, you should make sure you understand both notations
fully and have thought sufficiently about their implications before
you make a decision. Don't take my (or anyone else's) word for it!

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

/tuning/topicId_20929.html#21250

> > At the moment, I am rather fascinated by the IMPLICATIONS of the
> > Johnston notation... the idea that somehow maybe a player could
> > adjust his performance by one of the various commas indicated
> > in the score.
> >
> > This would, of course, be a much more AURAL process than
> > writing some kind of cents notation. The thought of this has
> > me mesmerized at the moment.
> >
> > HOWEVER, I am not fully convinced that a performer could do this.
> > Maybe if the tempo were VERY slow, but how about in fast passages?
> >
> > Does anybody know the success of Ben Johnston in making this
> > happen with real, live instrumentalists?? Perhaps I need to
> > know more about Johnston.
>
>
> As I tried to emphasize in my previous post, Joe, Johnston has
> fairly good success with his method *because* the music he
> writes is permeated with vertical JI sonorities (o- and
> u-tonalities).
>
> Therefore, once the performers understand both the notation
> and the compositional style, it's pretty easy to get decent
> results. Of course, results vary widely depending on the
> performer...
>
> I met Johnston in New York in (IIRC) December 1996, when Eric
> Grunen and a small orchestra performed his _Chamber Symphony_.
> I attended the rehearsal, and, while I was unfamiliar with the
> score previously, as I looked at it, it seemed to me that
> most of the performers were navigating the intonational field
> quite admirably.

That's when I met him too... Unfortunately, I didn't know YOU at
that time! I remember that it was quite a nice piece, but I would
have no idea if the intonation was correct. Too bad about Eric
Grunen. That's a real good example of somebody with a lot of
ambition who spent above his means...

>>
>
> > I am STILL unconvinced that performers
> > could really use this system even if a composer could figure
> > it out. For me, I would need some kind of charts or more
> > systematic instruction in order to work practically with it.
> >
> > It's not like I'm "well, isn't this theory interesting... now on
to the next thing..." I would really want to understand it in order
to COMPOSE with it, not just regard it as a curiosity...
> >
> > The fact that Kyle Gann and David Doty BOTH use it extensively
> > augurs well for it as a system. HOWEVER, I am under the
> > impression that BOTH of them use it MOSTLY with ELECTRONIC
> > works! I could be wrong, but that is the impression I have.
> > I have NO idea how much practical experience they have had
> > with "real" performers with it...
>
>
> Please keep in mind the caveat I wrote above. Johnston's notation
> works very well for both Gann and Doty *because* they compose
> in JI, and that 5-limit basic scale really *is* a basic part of
> their musical vocabulary.
>
> Based on what I know of your own compositions, Joe, it seems to
> me that the Wolf/Monzo Pythagorean-based notation would work
> better for you. I've never known any of your music to prominently
> feature 5-limit JI. Based on the logical superiority of the
> Pythagorean-based approach, I think you would favor it.
>

It looks like I might... perhaps it would be best to go for
"Monzotone" rather than Wolf... since I am in more frequent
communication with you...

>>
> Here's a quick tutorial:
>
> Johnston's notation assumes that 7-tone 5-limit scale as its
> basis, which means that the 7 letter names with no other
> accidentals represent that scale: A B C D E F G.
>
> The sharps and flats in Johnston's system represent an alteration
> of any of those 7 notes by a 25:24. Thus, if C = 1/1, then
> C# = 25/24. If E = 5/4, then Eb = 6/5. Etc.
>
> The big problem I have is that, after 5, all of Johnston's
> accidentals logically and consistently represent a *prime-factor*
> intonational inflection, but he muddled this distinction when
> dealing with prime-factors 3 and 5. So the plusses and minuses
> (+/-) represent the 81:80 syntonic comma, but in a "weird"
> pattern than can really only be understood by invoking lattice
> visualizations.

whoaboy... This is where I want to get off the train...

>
> It would be better to have the plain letters A B C D E F G
> to represent *only* prime-factor 3, as it already does
> historically in our standard notation.
>
> Then, the sharps and flats would represent an inflection of
> 2187:2048 (== 3^7), the usual Pythagorean "chromatic semitone"
> or "apotome", again, as it already does historically in our
> standard notation.
>
> The two advantages to this:
>
> 1)
> Elimination of the presence of prime-factor 5 in the sharps
> and flats, so that they only indicate intonational adjustments
> involving prime-factor 3, and so they too are on the same footing
> as all other accidentals.
>
> 2)
> The presence of a plus or minus *always* indicates the presence
> of prime-factor 5. This puts the +/- accidentals on the same
> footing as all the others (for 7, 11, 13, etc.).
>

The Pythagorean basis seems entirely logical and even historically
based...

>
> In Johnston's notation, both the #/b *and* +/- accidentals
> *always* indicate an adjustment which includes *both*
> prime-factors 3 and 5, because his basic scale includes both.
> This is my primary difficulty with his system.
>

You're saying, then, that it's not entirely clear what's implied when
using either the #/b or +/- ??

I'm still not entirely getting this... but since you say it's
confusing, maybe it's SUPPOSED to be confusing... so I'm on the right
track...

> Johnston's approach is fairly easy to conceptualize IF THE
> RESOURCES REMAIN FAIRLY SMALL, which is not the case even
> in his own music.
>

Well, that's bad, isn't it. After all, Johnston is a person who, if
I understand correctly, took the complexities of Western music,
serialism, et al and applied microtonality to it... So one would
expect his music to be rather complex, no??

> The Pythagorean-based approach is even simpler to conceptualize,
> but *MUCH* simpler to extend, because of its linearity, and
> because of the rigid consistency of its "rules" in regard to
> the meaning of accidentals.
>
>
> The notation is adapted to the kind of music written by the
> composer who invented it. Same as with Partch. It works
> for Johnston, and for several other JI composers, but that
> doesn't automatically mean that it would be well-suited for
> you or anyone else.
>

Well, perhaps I shouldn't admit this, but I'm actually going the
other way around (for better or worse!). I'm interested in exploring
different tuning systems and ways of notation and feel that my music
and personal style can adapt to them...

Of course, there are some people who will feel I am just using the
"flavor of the month."...

>
> Fonville, John. 1991.
> "Ben Johnston's Extended Just Intonation: A Guide for
Interpreters."
> _Perspectives of New Music_, vol 29, no 2 [Summer], p 106-137.
>
>

I just went here to their website:

http://depts.washington.edu/pnm/

and have asked about ordering this back issue...

> Of course, the best way to understand all this is to vault over
> the theoretical articles and study the scores themselves.

Yes, but these scores are not going to make any sense if I don't have
a "key" to the notation, correct??

>
> Two recommendations:
>
> 1)
> The most easily available Johnston recording is his _4th Quartet_
> (variations on "Amazing Grace" - it goes up to 7-limit) on the
> Kronos Quartet CD _White Man Sleeps_. Big chunks of this
> score also appears in the following article:
>

_White Man Sleeps_ is I think what's going to happen if I ever "get a
grip" on this notation...

> Shinn, Randall. 1977.
> "Ben Johnston's Fourth String Quartet".
> _Perspectives of New Music_, vol 15 no 2 [Spring-Summer], p
145-173).
>
> Note that this piece is especially a good starting point for you
> because I give lattices of its scales in my book.
>
>
> 2)
> In my opinion, the most outstanding example of Johnston's ability
> and talent is his _6th Quartet_. A great analysis of this was
> given in:
>
> Elster, Steven. 1991.
> "A Harmonic and Serial Analysis of Ben Johnston's
> String Quartet No. 6".
> _Perspectives of New Music_, vol 29 no 2 [Summer], p 138-165.
>
> (Note that this is the same issue as the Fonville article.)
>

That's good, since that's the one I want to order!

>
> There, that should keep you busy studying for quite some time...
>

I really appreciate all your input on this, Monz. You're obviously a
real authority on all of this!!!

______ _____ _____ _
Joseph Pehrson

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21253

> --- In tuning@y..., jpehrson@r... wrote:
> >
> > SO, what's the 25:24 all about??
> >
> Hi Joseph.
>
> You may recall, from our off-list conversations a couple of weeks
> ago, me explaining to you how the 25:24 is the JI chromatic
semitone, hence Ben Johnston's chromatic symbols (#, b) indicate
raising or lowering by a 25:24. My new paper will clarify this
further . . .

This is gradually "sinking in..."

> > This is a problem, man, a problem...
>
> I don't see what the problem is. Who's forcing you to use
Johnston's notation rather than Wolf's notation for your own music?

Well, the point is that I should KNOW the Johnston AND MonzoWolf
system thoroughly as part of my xenharmonic studies... As my music
evolves and develops more in these directions, I will, thereby, know
the right kind of notation to use... So some of this is a bit in the
"R&D" area...

>Or you might eeven want to use 72-tET notation (if you're not
moving across vast stretches of the lattice)

I never had any trouble conceptializing 72-tET... and there is quite
a bit of PRACTICAL literature out about it. That's a LOT different,
from a practical standpoint, than these "comma based" systems, in MY
opinion...

>or cents notation . . . perfectly good notations when the final
>result desired is 11-limit JI harmony . . . and perhaps easier to
>train 12-tET-trained musicians with.
>

Well, of course, this is how I am feeling all along, but I still need
to explore...

________ ______ _____ _
Joseph Pehrson

Joe, I'll write an explanation of my notation next.
For now...

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_20929.html#21387

> I'm still not entirely getting this... but since you say it's
> confusing, maybe it's SUPPOSED to be confusing... so I'm on
> the right track...

Very funny. I had to laugh out loud! (really, I did...)

No, Joe, of course, to Johnston and many of his numerous students,
the notation is clear and convenient.

> ...
>
> > Johnston's approach is fairly easy to conceptualize IF THE
> > RESOURCES REMAIN FAIRLY SMALL, which is not the case even
> > in his own music.
> >
>
> Well, that's bad, isn't it. After all, Johnston is a person who, if
> I understand correctly, took the complexities of Western music,
> serialism, et al and applied microtonality to it... So one would
> expect his music to be rather complex, no??

Johnston's music is just about the *most* intellectually complex
that I've ever come across. You can get a sense of it by
reading the Elster analysis of the _6th Quartet_. This
complexity is a very big part of the reason why I wish *he*
used the more logical Pythagorean-based approach.

He simply chose for his basic scale what he thought would be
the most logical: the 5-limit JI diatonic major. In his opinion,
musicians who needed to get a grasp of 5-limit JI in order to
play his music correctly in tune would be able to understand
his extensions of the system best if the understood this scale
as its basis.

Wolf, Erlich, and I collectively disagree with this because we
prefer the logical consistency of the Pythagorean-based notation.

> > ...
> > The most easily available Johnston recording is his
> > _4th Quartet_ (variations on "Amazing Grace" - it goes up to
> > 7-limit) on the Kronos Quartet CD _White Man Sleeps_.
> > Big chunks of this score also appears in the following article:
> >
>
> _White Man Sleeps_ is I think what's going to happen if I ever
> "get a grip" on this notation...

You're a real hoot, Joe! (my sides are aching...)

_White Man Sleeps_ #1 and #5 [1985] are also pieces by Kevin
Volans that are decent.

And the CD features a couple of other goodies:

- Ives's _Scherzo: Holding Your Own_ [1903-14]

- the Kronos quartet arrangement of Ornette Coleman's
_Lonely Woman_ [1959]

- a very aggressive (which = very good, IMO) performance
of Bartók's _3rd Quartet_ [1927].

And for those keeping tabs on the topical relevance to this list:

Since the whole CD is played by string quartet, it's
_de facto_ at least kind of microtonal, but the Coleman
and Johnston are explicity so. Johnston's intonation I
understand, Coleman's I don't...

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

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21384

> --- In tuning@y..., jpehrson@r... wrote:
>
> > It seems that Wolf's notation gets a little bizarre in terms of
> > accidental usage, now?? It looks a little as though somebody fell
> > the wrong way on the typewriter....
>

> Isn't that equally the case with Johnston's notation?

Ummm. Not quite. I have the 1/1 article in front of me with the
Wolf accidentals. Have you seen it recently? One accidental looks a
little like a lasso with a line across it... looks like the New York
City Ballet logo... One is a flat with a question mark attached to it
on top -- that would NOT be very encouraging to the questioning
performer... There is a double-x that looks as though one had
several Martinis before writing it, and, worst of all for the
computer age, there is an "@" sign, that might make somebody want to
go do e-mail right when they were trying for the 23rd limit!

No, it's worse than Johnston's.

> > >
> > I will try to find the Perspectives article on Ben Johnston's
> > notation... although I am beginning to doubt that it is as
"direct" as these "improved" systems.
> >
> > The reason, though, that I am so interested in it is, obviously,
> > Johnston's stature as a composer and xenharmonic thinker and the
> > usage of such composers as Doty and Gann...
> >

> I don't think either of these men would compose any differently if
> they happened to use Helmholtz/Ellis/Wolf/Monzo notation rather
than Johnston's. They both have a clear grasp (and have provided the
two clearest explanations) of what JI is all about. JI is about sound
(at least to many of us -- i know Dante Rosati has a different
> philosophy, so we should remain respectful of that). Perhaps some
> serial composition is more about notes on a page than about sound.
> But to these two men, JI is about sound. These men would simply
write in ratios if that were a useful notation. It wouldn't affect
the music they wrote.
>

Oh sure, Paul... Actually, I got it the first time you posted this on
the list. It is clear that these composers *do* know what they are
doing...

> By the way, did you catch David Doty's reply to you on this list?
>

Absolutely! And there was not much to say, since I was not intending
to deprecate either him or Kyle Gann...

I was a little confused in my understanding, since the implication
for me after I read the Doty article was that somehow these composers
were writing things down on the staff and that actually, because of
the Pythagorean basis of the staff, they weren't really coming out
they way they intended.

That was my initial and "confused" understanding, since I see now
that these composers would certainly know when they hit the just
ratios they wanted, REGARDLESS of the notational system. At least
this is what I think you are saying, and how I presently understand
it...

It STILL doesn't help me write in the system, though...

> But you still haven't said much about how you expect yout notation
to be used. Who's going to be reading it?

Nobody. Nobody that *I* can train, THAT'S for certain. We get TWO
rehearsals in New York and then go on stage. If I want a REALLY UGLY
performer, I'll just give her a Johnston "quick course."

>How can you best make use of the training they already have?

Not with this system, for GD sure... MAYBE I could get them to do
72-tET... at least the BASIC 12-tET system is in there... With a
little training MAYBE... at least it's in the realm of possibility...
but THIS. Forgetaboutit. It's great when the performer is a
synthesizer, and it looks mighty nice on paper... rather cryptic,
too, for those interested in a "cult item..."

>How can you minimize the number of new concepts they have to learn?

Not by using this GD thing!

Those are the questions i would be asking, rather than worrying
about anyone's "stature".
>

Good one, Paul! There are a lot of pretty stupid people around with
"stature!" And sometimes they even make "statues" out of them...

Well, those questions practically answer themselves... Not too much
thought required there, Paul!

> Anyway, here are some just major and minor triads in Johnston's
> notation:
>
> Bb- D- F (major)
>
> Bb D F+ (major)
>
> B- D- F# (minor)
>
> B D F#+ (minor)
>
> C E G (major)
>
> C Eb G (minor)
>
> D- F# A (major)
>
> D F#+ A+ (major)
>
> D- F A (minor)
>
> D F+ A+ (minor)
>
>
> Now the same triads in Wolf's notation:
>
> Bb D- F (major)
>
> Bb+ D F+ (major)
>
> B- D F#- (minor)
>
> B D+ F# (minor)
>
> C E- G (major)
>
> C Eb+ G (minor)
>
> D F#- A (major)
>
> D+ F# A+ (major)
>
> D- F A- (minor)
>
> D F+ A (minor)
>
>
> Note that in Wolf's notation, the root and fifth always have the
same commatic accidental applied to them; the minor third has an
extra + (or one less -), and the major third has an extra - (or one
less +). These conventions are easy to remember since the alterations
are measured from the Pythagorean basis of Wolf's notation.
>

Thanks, Paul... this is really a great comparison... actually the
first time I've really understood ANYTHING about EITHER of these
systems.

I'm assuming that there is more than one major or minor triad with
the
same letter names because a different FUNDAMENTAL is implied (??) I
guess so... That's kind of confusing to begin with... for BOTH
systems.

> In Johnston's notation, there is no consistent pattern to the way
the triads are notated. You have to look at a lattice diagram to know
> where you are.

That's just terrible. Back to composing... I'm sick of this stuff.

>Even if you take for granted that all triads in a
> score are going to sound just, reckoning the melodic intervals from
> one to the other is still going to require you to stop and look at
> the lattice (until you have the process memorized).

Oh... please get me an asperin.

>
> Of course, Joseph, you should make sure you understand both
notations fully and have thought sufficiently about their
implications before you make a decision. Don't take my (or anyone
else's) word for it!

Well, I've made a decision... bye to them. But, I'm still going to
study them since I'm a happy masochist...

_______ _______ ____ _
Joseph Pehrson

Dear Monzo,

What don't you understand about Ornette Coleman's tuning?

Johnny Reinhard

--- In tuning@y..., Afmmjr@a... wrote:

/tuning/topicId_20929.html#21428

> Dear Monzo,
>
> What don't you understand about Ornette Coleman's tuning?
>
> Johnny Reinhard

Hi Johnny. Well, I *do* know that he calls his system
"harmelodic" (sp?), and that it has something to do with
the interchangeability of clefs. But I don't know anything
specific about the tuning.

I seem to recall that we've gone over this here before,
within the last year or so, but I don't remember any details.
Feel free to expound.

-monz

--- In tuning@y..., jpehrson@r... wrote:>
>
>
> I never had any trouble conceptializing 72-tET... and there is quite
> a bit of PRACTICAL literature out about it. That's a LOT
different,
> from a practical standpoint, than these "comma based" systems, in
MY
> opinion...

One degree of 72-tET _is_ the syntonic comma in that system . . . so
I don't see how that's different.

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

> He simply chose for his basic scale what he thought would be
> the most logical: the 5-limit JI diatonic major. In his opinion,
> musicians who needed to get a grasp of 5-limit JI in order to
> play his music correctly in tune would be able to understand
> his extensions of the system best if the understood this scale
> as its basis.
>
> Wolf, Erlich, and I collectively disagree with this because we
> prefer the logical consistency of the Pythagorean-based notation.

I also disagree because I don't think the "5-limit JI diatonic major"
scale is even suitable for very much extant music.

--- In tuning@y..., jpehrson@r... wrote:
> --- In tuning@y..., PERLICH@A... wrote:
>
> /tuning/topicId_20929.html#21384
>
>
> > --- In tuning@y..., jpehrson@r... wrote:
> >
> > > It seems that Wolf's notation gets a little bizarre in terms of
> > > accidental usage, now?? It looks a little as though somebody
fell
> > > the wrong way on the typewriter....
> >
>
>
> > Isn't that equally the case with Johnston's notation?
>
>
> Ummm. Not quite. I have the 1/1 article in front of me with the
> Wolf accidentals. Have you seen it recently?

No -- I guess I just assumed that Wolf pretty much used the same
symbols as Johnston.

> One accidental looks a
> little like a lasso with a line across it... looks like the New
York
> City Ballet logo... One is a flat with a question mark attached to
it
> on top -- that would NOT be very encouraging to the questioning
> performer... There is a double-x that looks as though one had
> several Martinis before writing it, and, worst of all for the
> computer age, there is an "@" sign, that might make somebody want
to
> go do e-mail right when they were trying for the 23rd limit!
>
> No, it's worse than Johnston's.

OK -- I was unaware of that.

>
> >How can you best make use of the training they already have?
>
> Not with this system, for GD sure... MAYBE I could get them to do
> 72-tET... at least the BASIC 12-tET system is in there... With a
> little training MAYBE... at least it's in the realm of
possibility...
> but THIS. Forgetaboutit. It's great when the performer is a
> synthesizer, and it looks mighty nice on paper... rather cryptic,
> too, for those interested in a "cult item..."

Johnny claims to get good results with cents notation . . . would you
ever consider that?

>
> Thanks, Paul... this is really a great comparison... actually the
> first time I've really understood ANYTHING about EITHER of these
> systems.

Glad I could help.
>
> I'm assuming that there is more than one major or minor triad with
> the
> same letter names because a different FUNDAMENTAL is implied (??)

Well yes . . . a syntonic comma apart. Let's notate the C-F-Dm-G-
C "comma pump" progression using both notations:

Johnston:

G A A B- C-
E F F G- G-
C C D- D- E-

Wolf:

G A- A- B-- C-
E- F F G- G-
C C D- D- E--

Does that make things any clearer?

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21443

> --- In tuning@y..., "monz" <MONZ@J...> wrote:
>
> > He simply chose for his basic scale what he thought would be
> > the most logical: the 5-limit JI diatonic major. In his opinion,
> > musicians who needed to get a grasp of 5-limit JI in order to
> > play his music correctly in tune would be able to understand
> > his extensions of the system best if the understood this scale
> > as its basis.
> >
> > Wolf, Erlich, and I collectively disagree with this because we
> > prefer the logical consistency of the Pythagorean-based notation.
>
> I also disagree because I don't think the "5-limit JI diatonic
> major" scale is even suitable for very much extant music.

Good point, Paul. Of course, this scale *does* work well as
a basis for Johnston's own music... but using it *as* a basis
makes the notational operations overall more complex than they
need to be.

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

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

> Of course, this scale *does* work well as
> a basis for Johnston's own music... but using it *as* a basis
> makes the notational operations overall more complex than they
> need to be.

Hmm . . . I recall looking at a score of Johnston's String Quartet
#4, and banging by head on the wall each time I had to figure out
what the next chord was . . . it seemed that even for Johnston's own
music, a HEWM notation would be much more comprehensible . . .

Here is Johnston on his own notation (quoted from
http://www.paristransatlantic.com/magazine/interviews/johnston.html):

"From the very beginning, my whole aim was to keep my notation as
close to ordinary usage as I could; in fact, for a long time I was
only dealing with extended triadic usage, that is, no prime overtones
higher than five. The only extra symbols needed are the plus and
minus, which indicate the syntonic comma.
The syntonic comma is the difference between the whole step in-
between DO and RE and the whole step in-between RE and MI. The first
is a Pythagorean second, a 9:8 relation. The other is a 10:9
relation, which is the gap between a just major third and the
Pythagorean whole-step; the difference is 81:80. That difference must
be indicated, because the lack of care about it is what makes playing
in just intonation so difficult."

Now this is really ironic -- granting Johnston's rationalization of
DO, RE, and MI, Johnston's actual notation notates these notes C, D,
and E, with no further accidentals. He says "that difference must be
indicated", but in his own notation, that difference is concealed!

Next:

"Hindemith talks about this problem in The Craft of Musical
Composition, but as a violist, not as a composer. He speaks about the
necessity to conceal the comma in playing. String players will
naturally try to play Mozart, or any other triadic music, with just
triads. But in doing so, they run into the problem of the comma. Now,
they won't make the mistake of playing LA-flat rather than SOL-sharp;
that's an egregious mistake; it's 40 cents off, nobody would do that.
But twenty cents off, maybe. The comma is only about 22 cents.
Indeed, if they were to play the progression I-VI-II-V-I over and
over again, they would move by common tone, dropping a comma every
time. After five repetitions, they would be a half-tone flat!
So the pluses and minuses are necessary; I just added those to the
ordinary notation."

Here it's uncertain what Johnston means -- are the pluses and minuses
necessary to prevent the drift, or to accurately notate it? In the
latter case, well some of Johnston's compositions purposely use
drift . . . but in Mozart, I don't think we'd want it (this is tonal
music, after all, which over the course of a piece modulates from the
tonic and then returns to it) because, in my opinion, good string
players playing just triads would hide the comma not by putting it
all in one place melodically, but by dividing it into smaller melodic
chunks that pass by unnoticed.

"Then when I wanted to go an extra step, I asked myself ­ what
interval is the seventh partial closest to? It's closest to a minor
seventh, lowered by 49 cents."

Now of course Johnston means the 9:5 minor seventh and not the 16:9
minor seventh. But why? Even the 5-limit JI diatonic scale that forms
the basis of his notation has three of the latter and only two of the
former.

"Do these terms get in your way, major third, perfect fifth..."

"Not really. You just have to realize that all terminology is a means
to an end. And if you lose sight of the end, the terminology's not
going to save you!"

Amen!

Oops . . . I blurred some thoughts. This:

"but in Mozart, I don't think we'd want it (this is tonal
music, after all, which over the course of a piece modulates from the
tonic and then returns to it) because, in my opinion, good string
players playing just triads would hide the comma not by putting it
all in one place melodically, but by dividing it into smaller melodic
chunks that pass by unnoticed."

should read:

"but in Mozart, I don't think we'd want drift (this is tonal
music, after all, which over the course of a piece modulates from the
tonic and then returns to it); and I don't think Johnston's (or HEW)
notation would be a good way of addressing this, because, in my
opinion, good string players playing just triads would hide the comma
not by putting it all in one place melodically, but by dividing it
into smaller melodic chunks that pass by unnoticed."

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21455

> --- In tuning@y..., "monz" <MONZ@J...> wrote:
>
> > Of course, this scale *does* work well as
> > a basis for Johnston's own music... but using it *as* a basis
> > makes the notational operations overall more complex than they
> > need to be.
>
> Hmm . . . I recall looking at a score of Johnston's String
> Quartet #4, and banging by head on the wall each time I had
> to figure out what the next chord was . . . it seemed that even
> for Johnston's own music, a HEWM notation would be much more
> comprehensible . . .

Exactly the point I was making when I wrote to Joe that "I
always end up cursing it [Johnston's notation] when I'm working
on his music".

The only way I can make a MIDI-file of his music is to have
pages of lattices in front of me, with both his notation and
mine... and, of course, the MIDI pitch-bend values too.
It's aggravating.

> > [Johnston:]
> > The syntonic comma is the difference between the whole step
> > in-between DO and RE and the whole step in-between RE and MI.
> > The first is a Pythagorean second, a 9:8 relation. The other
> > is a 10:9 relation, which is the gap between a just major
> > third and the Pythagorean whole-step; the difference is 81:80.
> > That difference must be indicated, because the lack of care
> > about it is what makes playing in just intonation so difficult."
>
> Now this is really ironic -- granting Johnston's rationalization
> of DO, RE, and MI, Johnston's actual notation notates these
> notes C, D, and E, with no further accidentals. He says "that
> difference must be indicated", but in his own notation, that
> difference is concealed!

Yup - I agree. To me (and you too, I'm sure) it's perfectly
clear what he's doing. The difference is not a part of his
basic-scale notation, so it has to be indicated when it *does*
pop up elsewhere. But we both agree that the 81:64 should
have been the unaltered "major 3rd", and 5:4 ought to be the
one with the minus sign.

>
> Here it's uncertain what Johnston means -- are the pluses and
> minuses necessary to prevent the drift, or to accurately notate
> it?

Hmmm... he probably means both.

> In the latter case, well some of Johnston's compositions
> purposely use drift . . . but in Mozart, I don't think we'd
> want it (this is tonal music, after all, which over the course
> of a piece modulates from the tonic and then returns to it)
> because, in my opinion, good string players playing just triads
> would hide the comma not by putting it all in one place
> melodically, but by dividing it into smaller melodic chunks
> that pass by unnoticed.

I agree with you, Paul. Good players are probably more likely
to distribute the comma among several notes rather than put it
into one particular interval.

>
> > "Then when I wanted to go an extra step, I asked myself ­what
> > interval is the seventh partial closest to? It's closest to
> > a minor seventh, lowered by 49 cents."
>
> Now of course Johnston means the 9:5 minor seventh and not the
> 16:9 minor seventh. But why? Even the 5-limit JI diatonic scale
> that forms the basis of his notation has three of the latter
> and only two of the former.

It has to do with the meaning of the flat symbol as a 24:25.
If you start out with his basic 7-tone 5-limit scale, and
consistently apply his intonational rules to the appearance
of sharps and flats, you end up with a plain Bb that is a 9:5,
and 16:9 ends up as Bb-.

That's why the intonational adjustment for 7:4 is calculated
from 9:5 instead of 16:9. The adjustment between 16:9 and
7:4 requires not only the addition of a "7" accidental, but
also the loss of a minus sign. The adjustment between 9:5
and 7:4 requires only the addition of a "7".

Yet another reason why HEWM works better.

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

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

/tuning/topicId_20929.html#21459

> --- In tuning@y..., PERLICH@A... wrote:
>
> > > [Ben Johnston:]
> > > "Then when I wanted to go an extra step, I asked myself ­what
> > > interval is the seventh partial closest to? It's closest to
> > > a minor seventh, lowered by 49 cents."
> >
> > Now of course Johnston means the 9:5 minor seventh and not the
> > 16:9 minor seventh. But why? Even the 5-limit JI diatonic scale
> > that forms the basis of his notation has three of the latter
> > and only two of the former.
>
>
> It has to do with the meaning of the flat symbol as a 24:25.
> If you start out with his basic 7-tone 5-limit scale, and
> consistently apply his intonational rules to the appearance
> of sharps and flats, you end up with a plain Bb that is a 9:5,
> and 16:9 ends up as Bb-.
>
> That's why the intonational adjustment for 7:4 is calculated
> from 9:5 instead of 16:9. The adjustment between 16:9 and
> 7:4 requires not only the addition of a "7" accidental, but
> also the loss of a minus sign. The adjustment between 9:5
> and 7:4 requires only the addition of a "7".

I should have added that *this* is precisely what I was referring
to when I said in my long response to Joe that "the problemmatic
2-dimensional basic scale is replicated in every other dimension".

So it doesn't stop here: the same kinds of debatable assignments
of +/- happen again, in Johnston's own music, with prime-factors
11, 13, 17, 19, 23, 29, and 31, because he has composed in up
to 31-prime-limit.

It's all too complicated for me, and if I'm not using HEW, then
I'll at least stick with M, thank you.

(And 72-EDO for practical and/or supplementary purposes.)

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

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21441

> --- In tuning@y..., jpehrson@r... wrote:>
> >
> >
> > I never had any trouble conceptializing 72-tET... and there is
quite a bit of PRACTICAL literature out about it. That's a LOT
> different, from a practical standpoint, than these "comma based"
systems, in MY opinion...
>
> One degree of 72-tET _is_ the syntonic comma in that system . . .
so I don't see how that's different.

Hi Paul...

Well, actually, I didn't know that! So, they are, technically BOTH
comma-based systems. However, 72-tET is, compared to the Johnston
system, easy to play and notatate... since 12-tET is contained in it
and there are simply 6 inflections for each basic step, correct??

______ _____ _____ _____
Joseph Pehrson

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21446

> > 72-tET... at least the BASIC 12-tET system is in there... With a
> > little training MAYBE... at least it's in the realm of
> possibility... but THIS. Forgetaboutit. It's great when the
performer is a synthesizer, and it looks mighty nice on paper...
rather cryptic,
> > too, for those interested in a "cult item..."
>

> Johnny claims to get good results with cents notation . . . would
you ever consider that?

Of course... er... that's what I USE! I'm just "messing around"
trying to figure out this other stuff to increase my "general
understanding..." hopefully...

>
> > I'm assuming that there is more than one major or minor triad with
> > the same letter names because a different FUNDAMENTAL is implied
(??)
>
> Well yes . . . a syntonic comma apart. Let's notate the C-F-Dm-G-
> C "comma pump" progression using both notations:
>
> Johnston:
>
> G A A B- C-
> E F F G- G-
> C C D- D- E-
>
> Wolf:
>
> G A- A- B-- C-
> E- F F G- G-
> C C D- D- E--
>
> Does that make things any clearer?

OHMYGOD... the light is striking! In the Wolf notation you can
actually SEE that the major thirds are smaller than the Pythagorean
and the minor thirds are larger! It's consistent with that
throughout!

Although I believe it STILL would be hard for a conventional
performer to learn at least it MAKES SENSE... systematically...

This is the best post so far!

_______ ______ _____ ____
Joseph Pehrson

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21455

> Here is Johnston on his own notation (quoted from
>
http://www.paristransatlantic.com/magazine/interviews/johnston.html):
>

Thanks for posting this, Paul... it was very interesting...

_____ ___ ______ ____
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
>
> Well, actually, I didn't know that! So, they are, technically BOTH
> comma-based systems. However, 72-tET is, compared to the Johnston
> system, easy to play and notatate... since 12-tET is contained in
it
> and there are simply 6 inflections for each basic step, correct??

Yes -- and what's more, these inflections are immediately correlated
with the prime-factors that you're using -- just as in the HEWM
notation!

This amazing fact stems from the fact that the unaltered pitches in
72-tET notation, 12-tET, play the role of the Pythagorean tuning in
HEWM. And of course, 12-tET is very close to Pythagorean tuning.

The entire 72-tET notation is simply 6 interlocking 12-tET systems.

Every factor of 5 you put in the numerator (or move up in the
lattice), you have to use the 12-tET system 1/72 octave lower (or
1/12 tone lower) than standard 12-tET. And the reverse for every
factor of 5 you put in the denominator. So the 1/12 tone down
indicators play the role of the -, and 1/12 tone up indicators play
the role of the +, in 72-tET notation.

Every factor of 7 you put in the numerator, you have to use the 12-
tET system 2/72 octave lower (or 1/6 tone lower) than standard 12-tET.

Every factor of 11 you put in the numerator, you have to use the 12-
tET system 3/72 octave lower (or 1/4 tone lower) than standard 12-tET.

What's really great is that these accidentals don't pile up. In HEWM
notation you might have to notate a note with various 5-based, 7-
based, and 11-based accidentals. In 72-tET notation, you're always in
one of the 6 12-tET systems . . . no sweat!

Accuracy: The entire 11-limit Tonality Diamond (Partch's 29 "primary
ratios" in the 2/1) is represented with a maximum error of 4 cents.
Well within the "subconscious adjustment range" within which players
can seek to eliminate beats, if so instructed.

Uniqueness: Every 11-limit interval (I mean odd limit, of course) is
represented by a _different_ 72-tET interval.

Consistency: You have consistency through the 17-limit -- so there
will be no possible confusion when notating, say, big 17-limit
otonalities.

Hence I think _most_ JI music can be adequately, and very easily,
notated in 72-tET. The exceptions would be

1) If you had a progression which "pumped" one of the commas that
vanish in 72-tET, such as the 224:225. This progression would not
drift in 72-tET, even though it would in JI (by about 7 cents each
time you repeated it).

2) If you wanted to use 19-limit or higher sonorities in an intricate
manner with accurate intonation . . . the notational inconsistencies
could cause practical difficulties here.

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21486

> --- In tuning@y..., jpehrson@r... wrote:
> >
> > Well, actually, I didn't know that! So, they are, technically
BOTH comma-based systems. However, 72-tET is, compared to the
Johnston system, easy to play and notatate... since 12-tET is
contained in it and there are simply 6 inflections for each basic
step, correct??
>
> Yes -- and what's more, these inflections are immediately
correlated with the prime-factors that you're using -- just as in the
HEWM notation!
>
> This amazing fact stems from the fact that the unaltered pitches in
> 72-tET notation, 12-tET, play the role of the Pythagorean tuning in
> HEWM. And of course, 12-tET is very close to Pythagorean tuning.
>
> The entire 72-tET notation is simply 6 interlocking 12-tET systems.
>

This is truly incredible! I'm not sure I entirely understand why it
works out this way... Why would an extended Pythagorean chain produce
all those other limits??

_______ ______ _____ _______
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

/tuning/topicId_20929.html#21478

> --- In tuning@y..., PERLICH@A... wrote:
>
> /tuning/topicId_20929.html#21446
>
> > > I'm assuming that there is more than one major or minor
> > > triad with the same letter names because a different
> > > FUNDAMENTAL is implied (??)
> >
> > Well yes . . . a syntonic comma apart. Let's notate the
> > C-F-Dm-G-C "comma pump" progression using both notations:
> >
> > Johnston:
> >
> > G A A B- C-
> > E F F G- G-
> > C C D- D- E-
> >
> > Wolf:
> >
> > G A- A- B-- C-
> > E- F F G- G-
> > C C D- D- E--
> >
> > Does that make things any clearer?
>
>
> OHMYGOD... the light is striking! In the Wolf notation you
> can actually SEE that the major thirds are smaller than the
> Pythagorean and the minor thirds are larger! It's consistent
> with that throughout!

Exactly! This is the point I've been hammering home.

Joe, since you like Paul's example of the "comma pump" here so
much, I'll refer back to the post you wrote right before this
one, in which Paul explained that the step-size of 72-tET is
also the size of the comma in that system, and tie it into this.

I think at this point it would probably be a good idea to
illustrate the pitches in Paul's example on a lattice.
This is in Wolf's version of HEWM notation:

E--...B--
/ \ / \
/ \ / \
C-....G-....D-....A-....E-
\ / \ / \
\ / \ / \
F.....C.....G

Note that you were able to easily follow Paul's line of argument
and understand exactly what he illustrated in HEWM notation
*without* having to refer to a lattice!

Unless you've got an incredibly good memory for spatial
relationships, this would be much more difficult in Johnston's
system.

Since it's now easy for you to see how the +/- accidentals
in HEWM notation designate commatic differences, all I have
to point out is that in 72-tET it works pretty much the same
way. The minus sign which indicates that a 12-tET pitch is
to be flattened by 1/12-tone would give a set of pitches notated
just like the middle row in this lattice, and in the most
common context, these notes would also have the same harmonic
or scalar function as the JI notes in HEWM notation. This
is what Paul meant when he said that +/- also designates the
comma in 72-tET.

The difference lies in the increasing exponents: that is,
when we get to the ratios in the top row of this lattice,
which have 5^2 as a factor, 72-tET doesn't have to use "--"
because it already has "<" (or an equivalent, depending on
whose version you're using) to represent a flattening by
1/6-tone. So in 72-tET (or 72-EDO, as I prefer to call it),
the lattice looks nearly identical, like this:

E<....B<
/ \ / \
/ \ / \
C-....G-....D-....A-....E-
\ / \ / \
\ / \ / \
F.....C.....G

Here's another interesting point to note. 72-tET uses
"<" to represent the basic septimal deviation from 12-tET
(~31 cents in JI, 33-&-1/3 cents in 72-tET *tuning*), so
that the 7th harmonic of "C" (i.e., the 7:4 ratio above 1/1)
would be notated "Bb<". Well, check this out...

Let's make a lattice illustrating the 5-limit JI "augmented 6th",
225:128 (= ~977 cents), which as Fokker demonstrated is only a
225:224 (~8 cents) wider than 7:4 :

In HEWM JI notation:

G#--..D#--..A#--
/ \ / \ /
/ \ / \ /
E-....B-....F#-
/ \ / \ / \
/ \ / \ / \
C.....G.....D.....A

In 72-tET notation:

G#<...D#<...A#<
/ \ / \ /
/ \ / \ /
E-....B-....F#-
/ \ / \ / \
/ \ / \ / \
C.....G.....D.....A

72-tET notation uses the same letter-names and #/b accidentals
as 12-tET, with all the same enharmonic equivalences. So
guess what? In 72-tET, A#< and Bb< are enharmonically equivalent,
that is, they are presumed to be the *same pitch*.

(I say "presumed" because in actual practice the composer may
intend for the players to use 72-tET as a rough guide and adjust
their intonation slightly to produce actual JI pitches.)

So a notational complication is done away with, because the
225:224 "vanishes".

Of course, this is not the case in HEWM notation, because
HEWM gives every prime-factor a distinct accidental. So if
the particular version of HEWM is using < and > to represent
an intonational inflection of 7^1 and 7^-1, respectively,
then the "harmonic 7th" is notated as Bb<, while (as can be
seen in the first of the two lattices above) the 5-limit
"augmented 6th" is notated as A#--.

Each of the two notations thus handles situations like this
in their own way. The way I see it, there are advantages and
disadvantages to both, so it's nice to be familiar with both
of them, to be able to use whichever one seems better for a
given compositional context... or even to combine them.

In my version of HEWM, with the numbers as accidentals, one
may combine both, as I explained in my other post, so that
the "harmonic 7th" may be written 7^1 Bb<, and the "augmented
6th" as 5^2 A#<. So in this case, the 72-tET Bb< or A#< gives
the performer a quick indication of the general pitch of the
note as "Bb minus 1/6-tone", and the numerical accidental
literally fine-tunes it as ~969 or ~977 cents respectively.

But of course, as with any JI composer, this fine-tuning is
meant to be understood and achieved by ear more than by eye.
The presence of a 5 in the accidental will tell the performer
that it's supposed to be a sweet 5:4 "major 3rd" relation to
another note in the chord, and the presence of 7 will tell
him/her that it's supposed to be a 7:4 relation to some note.

>
> Although I believe it STILL would be hard for a conventional
> performer to learn at least it MAKES SENSE... systematically...

On the contrary, the Boston-area microtonalists who have learned
the 72-tET notation from Ezra Sims and Maneri/vanDuyne have
demonstrated that it's easy to comfortably navigate the commatic
distinctions represented by 72-tET.

You have a copy of the Maneri/vanDuyne book, don't you?
Work your way thru just the first few sections, and you'll
be surprised at how quickly and easily you can grasp 72-tET.

As I said in my other post, sometimes I use 1200-EDO as a
supplement to HEWM instead of 72-EDO, as in my transcriptions
of Partch's _The Intruder_ and Johnson's _Drunken Hearted Man_.
So I mean no offense here to Johnny.

But in my experience, once you get used to 72-EDO, it seems
to seductively claim your attention. I know that these days
when I get to writing down the microtonal music I hear in my
mind, the first definite pinpointing of pitch usually happens
in 72-EDO, and the other intonational refinements come later.

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

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21486

> [A great explanation of good points about 72-tET notation.]

Thanks, Paul - this is really terrific!

Joe [Pehrson],

The post I just wrote (with the lattice diagrams of the comma
pump) says a lot of the same stuff Paul says here but in a
different way, so I think the two posts together give a good
overview of the value of 72-tET for notational purposes, and
how it compares to HEWM.

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

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21486

> This amazing fact stems from the fact that the unaltered pitches in
> 72-tET notation, 12-tET, play the role of the Pythagorean tuning in
> HEWM. And of course, 12-tET is very close to Pythagorean tuning.
>
> The entire 72-tET notation is simply 6 interlocking 12-tET systems.
>
> Every factor of 5 you put in the numerator (or move up in the
> lattice), you have to use the 12-tET system 1/72 octave lower (or
> 1/12 tone lower) than standard 12-tET. And the reverse for every
> factor of 5 you put in the denominator. So the 1/12 tone down
> indicators play the role of the -, and 1/12 tone up indicators play
> the role of the +, in 72-tET notation.
>
> Every factor of 7 you put in the numerator, you have to use the 12-
> tET system 2/72 octave lower (or 1/6 tone lower) than standard 12-
tET.
>
> Every factor of 11 you put in the numerator, you have to use the 12-
> tET system 3/72 octave lower (or 1/4 tone lower) than standard 12-
tET.
>
> What's really great is that these accidentals don't pile up. In
HEWM notation you might have to notate a note with various 5-based, 7-
> based, and 11-based accidentals. In 72-tET notation, you're always
in one of the 6 12-tET systems . . . no sweat!
>
> Accuracy: The entire 11-limit Tonality Diamond (Partch's
29 "primary ratios" in the 2/1) is represented with a maximum error
of 4 cents.
> Well within the "subconscious adjustment range" within which
players can seek to eliminate beats, if so instructed.
>
> Uniqueness: Every 11-limit interval (I mean odd limit, of course)
is represented by a _different_ 72-tET interval.
>
> Consistency: You have consistency through the 17-limit -- so there
> will be no possible confusion when notating, say, big 17-limit
> otonalities.
>

Hi Paul...

You've shown "adjustments" in 72-tET at the 5, 7, and 11 limit...but
how do you get 17 again??

_________ _____ ______
Joseph Pehrson

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

/tuning/topicId_20929.html#21502

> >
> > OHMYGOD... the light is striking! In the Wolf notation you
> > can actually SEE that the major thirds are smaller than the
> > Pythagorean and the minor thirds are larger! It's consistent
> > with that throughout!
>
>
> Exactly! This is the point I've been hammering home.
>
> Joe, since you like Paul's example of the "comma pump" here so
> much, I'll refer back to the post you wrote right before this
> one, in which Paul explained that the step-size of 72-tET is
> also the size of the comma in that system, and tie it into this.
>

Thanks so much, Monz, for these lattices... and they are, of course,
also very clear.... I have a better understanding of this stuff now
than I ever did.

(Well, I didn't know ANYTHING about it right at the beginning...)

> Since it's now easy for you to see how the +/- accidentals
> in HEWM notation designate commatic differences, all I have
> to point out is that in 72-tET it works pretty much the same
> way. The minus sign which indicates that a 12-tET pitch is
> to be flattened by 1/12-tone would give a set of pitches notated
> just like the middle row in this lattice, and in the most
> common context, these notes would also have the same harmonic
> or scalar function as the JI notes in HEWM notation. This
> is what Paul meant when he said that +/- also designates the
> comma in 72-tET.
>

Yes, I understand now that every 72-tET step is a comma...

> In HEWM JI notation:
>
> G#--..D#--..A#--
> / \ / \ /
> / \ / \ /
> E-....B-....F#-
> / \ / \ / \
> / \ / \ / \
> C.....G.....D.....A
>
>
> In 72-tET notation:
>
> G#<...D#<...A#<
> / \ / \ /
> / \ / \ /
> E-....B-....F#-
> / \ / \ / \
> / \ / \ / \
> C.....G.....D.....A
>
>
> 72-tET notation uses the same letter-names and #/b accidentals
> as 12-tET, with all the same enharmonic equivalences. So
> guess what? In 72-tET, A#< and Bb< are enharmonically equivalent,
> that is, they are presumed to be the *same pitch*.
>
> (I say "presumed" because in actual practice the composer may
> intend for the players to use 72-tET as a rough guide and adjust
> their intonation slightly to produce actual JI pitches.)
>
> So a notational complication is done away with, because the
> 225:224 "vanishes".
>

OK... so does that mean that the 225:224 in 72-tET is the "unison
vector??" Is that the "Periodicity Block" in action (??)

> > Although I believe it STILL would be hard for a conventional
> > performer to learn at least it MAKES SENSE... systematically...
>
>
> On the contrary, the Boston-area microtonalists who have learned
> the 72-tET notation from Ezra Sims and Maneri/vanDuyne have
> demonstrated that it's easy to comfortably navigate the commatic
> distinctions represented by 72-tET.
>

Monz, I think if you look back at the original post, I was referring
to Monzowolfellholtz and NOT 72-tET... I have *NO* questions about
the practicality of 72-tET... it's being proven every day in action!

(BTW -- just kidding about "Monzowolfellholtz"--- I just like the
word!)

________ ______ ______ ____
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
>
> This is truly incredible! I'm not sure I entirely understand why
it
> works out this way... Why would an extended Pythagorean chain
produce
> all those other limits??
>
It doesn't! In 72-tET, the Pythagorean chain is the same as in 12-
tET -- it closes after 12 fifths. It takes 6 _separate_ Pythagorean
chains to get you all 72 notes. And fortuitously, starting
from "regular" 12-tET representing Pythagorean ratios, the 5 extra
chains are obtained by, in ascending order

1) (1/12 tone up ) dividing ratios by 5
2) (1/6 tone up ) dividing ratios by 7
3) (1/4 tone up ) multiplying or dividing ratios by 11
4) (1/6 tone down) multiplying ratios by 7
5) (1/12 tone down) multiplying ratios by 5

This makes it _immediately_ apparant how to notate any interval from
any other interval.

For example, let's say we want to notate an 11/10 above C. Well, you
know (hopefully) that it's somewhere between C# and D. But where?
Well, the factor of 11 in the numerator takes you down a quarter-
tone, and the factor of 5 in the denominator takes you up a twelfth-
tone. So you're down a sixth-tone from the 12-tET system that
included the original note. So the answer is D-1/6-tone-flat!

You can learn this type of thinking very quickly . . . notating 11-
limit music becomes a breeze . . . I'm sure Ezra Sims can confirm
this and also testify to how accurately his music is generally
performed . ..

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21533

> In 72-tET, the Pythagorean chain is the same as in 12-
> tET -- it closes after 12 fifths. It takes 6 _separate_
> Pythagorean chains to get you all 72 notes. <etc. ...>

Joe (and others who are still mystified as to why 72-EDO
works so well in so many different contexts),

Study Paul's post well! This is a terrific concise explanation
of the virtues of 72-EDO notation *and* tuning.

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

--- In tuning@y..., jpehrson@r... wrote:
>
> You've shown "adjustments" in 72-tET at the 5, 7, and 11
limit...but
> how do you get 17 again??

Hi Joseph.

Firstly, note that the 17-limit, which includes ratios of 3, 5, 7, 9,
11, 13, 15 and 17, is not _uniquely_ expressed in 72-tET. 72-tET is
only _unique_ (and maximum errors are less than 4 cents) through the
11-limit. Beyond that, there are some ambiguous intervals, which only
a triadic or larger chordal context can clear up (and, the errors get
a little larger). Looking at the table at the bottom of Rick Tagawa's
homepage, http://sites.netscape.net/masanoritagawa/homepage, you'll
see a few of these ambiguities:

13:11 "=" 20:17
13:10 "=" 22:17

etc.

Anyhow, to answer your specific question, 72-tET's approximation of
the 17th harmonic actually is the 12-tET semitone. So the 17th
harmonic of C is C#, etc. This is only 5 cents off -- the 17th
harmonic is actually 105 cents (plus 4 octaves).

The 13th harmonic above C is approximated, in 72-tET, by A-1/3-tone-
flat, 50 steps of 72-tET, or 833.3 cents. This is 7 cents flat of the
true 13th harmonic.

So we can supplement our rules for factors of 5, 7, and 11 with rules
for 13 and 17. Every factor of 13 in the numerator moves you to the
12-tET system up 1/6-tone (or down 1/3-tone). Factors of 17 leave you
in the same 12-tET system you started in.

Once you hit the 19-limit, 72-tET is inconsistent. For example, the
best approximation of 8:19 is 90 steps of 72-tET; the best
approximation of 13:19 is 39 steps of 72-tET; but if you use these
approximations to construct an 8:13:19 triad, 8:13 would have to be
approximated by 90 - 39 = 51 steps of 72-tET -- but we've seen that
50 steps is a better approximation. Of course, 72-tET is such a fine
division of the octave that even such mis-approximations are still
pretty good, such that Ezra Sims has actually used 72-tET to notate
37-limit JI music. Whether this will work in general or not, of
course, depends on the particulars of the composition in question and
the musicians called upon to perform it.

Again, the stuff at the bottom of Rick's homepage should elaborate on
this and make it clearer . . .

--- In tuning@y..., jpehrson@r... wrote:
>
>
> OK... so does that mean that the 225:224 in 72-tET is the "unison
> vector??"

It is a unison vector -- just like 81:80 is a unison vector in 12-
tET, 19-tET, 31-tET, and all meantone temperaments . . .

>Is that the "Periodicity Block" in action (??)

Well, many periodicity blocks that have 72 tones have 225:224 as one
of the unison vectors . . . of course there are usually 2 or 3
others . . .

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21533
> >
> It doesn't! In 72-tET, the Pythagorean chain is the same as in 12-
> tET -- it closes after 12 fifths. It takes 6 _separate_ Pythagorean
> chains to get you all 72 notes. And fortuitously, starting
> from "regular" 12-tET representing Pythagorean ratios, the 5 extra
> chains are obtained by, in ascending order
>
> 1) (1/12 tone up ) dividing ratios by 5
> 2) (1/6 tone up ) dividing ratios by 7
> 3) (1/4 tone up ) multiplying or dividing ratios by 11
> 4) (1/6 tone down) multiplying ratios by 7
> 5) (1/12 tone down) multiplying ratios by 5
>
> This makes it _immediately_ apparant how to notate any interval
from any other interval.
>

Well, this is pretty amazing, if I'm understanding it correctly...

In other words, you're combining many of the REAL BASICS of tuning
theory in this scale:

You have chains of fifths, the basic building block of much of all
music, certainly all Western music, and you have the multiplication
and division by prime numbers associated with Just Intonation and
"limits..." So this scale "has it all!"

As a silly aside, this scale reminds me of the old "15-speed"
bicycles that used to be around (maybe they still are...).

There is a chain, and it could be placed on various sized "sprocket"
wheels to generate different speeds.

This was the image that immediately came to mind when I thought of
this scale! :)

________ _____ _____ ____
Joseph Pehrson

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21542

>
> Anyhow, to answer your specific question, 72-tET's approximation of
> the 17th harmonic actually is the 12-tET semitone.

Thanks, Paul... Actually, this struck me as rather humorous... but
it's late...

Thanks for the explanations!

_______ _____ _____ _
Joseph Pehrson

Hi Joseph!

The idea of 72-tET as a six-gear version of a 12-tET bike reminds me of
something similar we both learned when talking about your 19-tone just
scales:

171-tET (13-limit-consistent, 9-limit-ULTRA-accurate) as a nine-gear version
of 19-tET. Do you remember that? The difference is that 19-tET approximates
a chain of just minor thirds, not fifths, and that's kind of a weird lattice
direction for the bike gears to point in, if you know what I mean.

I found some others "bikes" among the super-accurate ETs (probably of
theoretical interest only and completely useless):

217-tET (19-limit-consistent, 13-limit-very-accurate) as a seven-gear
version of 31-tET. Here 31-tET approximates a chain of just major thirds or
just harmonic sevenths or septimal tritones. Three more weird gear
directions.

342-tET (11-limit-ULTRA-accurate) as a two-gear version of 171-tET, itself a
nine-gear version of 19-tET. Here's a bike you can really cruise the lattice
on, hugging those minor third grooves, of course.

494-tET (15-limit-ULTRA-accurate, 17-limit consistent) as a 26-gear version
of 19-tET, OR as a 19-gear version of 26-tET (which can be thought of as a
chain of just harmonic sevenths)

665-tET (3-limit-HYPER-accurate, 9-limit consistent) as a 35-gear version of
19-tET.

1178-tET (13-limit-ULTRA-accurate, 21-limit consistent) as a 62-gear version
of 19-tET.

1330-tET (3-limit-HYPER-accurate, 11-limit consistent) as a 2-gear version
of 665-tET, itself a 35-gear version of 19-tET.

Very accurate > 99.7% accuracy
ULTRA-accurate > 99.987% accuracy
HYPER-accurate > 99.9999999978% accuracy

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_20929.html#21567

> The idea of 72-tET as a six-gear version of a 12-tET bike
> reminds me of something similar we both learned when talking
> about your 19-tone just scales:
>
< <etc.>

Hey Paul, this post gave a terrific simple way of thinking
about some of these super-accurate/consistent EDOs.

Can you say anything about the mathematics of 318-EDO,
and explain why it fits my Pythagorean analysis of Aristoxenus's
tuning so well? You can find that near the end, under
"My Conclusions", at:
http://www.ixpres.com/interval/monzo/aristoxenus/318.htm

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

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

> Hey Paul, this post gave a terrific simple way of thinking
> about some of these super-accurate/consistent EDOs.
>
> Can you say anything about the mathematics of 318-EDO,

318-tET is six 53-tET near-Pythagorean circles. It's only consistent
through the 5-limit, and you'd never leave the first 53-tET circle
using any 5-limit ratios.

> and explain why it fits my Pythagorean analysis of Aristoxenus's
> tuning so well? You can find that near the end, under
> "My Conclusions", at:
> http://www.ixpres.com/interval/monzo/aristoxenus/318.htm

That link didn't work. Ironically, Aristoxenus's scales are usually
understood as subsets of 72-tET (q.v. Xenakis, etc.) You probably got
318-tET by observing the six circles of 12-tET in 72-tET and changing
each of the six Pythagorean circles from 12-tET to 53-tET, yes?

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21574

> --- In tuning@y..., "monz" <MONZ@J...> wrote:
>
> > Hey Paul, this post gave a terrific simple way of thinking
> > about some of these super-accurate/consistent EDOs.
> >
> > Can you say anything about the mathematics of 318-EDO,
>
> 318-tET is six 53-tET near-Pythagorean circles. It's only
> consistent through the 5-limit, and you'd never leave the
> first 53-tET circle using any 5-limit ratios.
>
> > and explain why it fits my Pythagorean analysis of Aristoxenus's
> > tuning so well? You can find that near the end, under
> > "My Conclusions", at:
> > http://www.ixpres.com/interval/monzo/aristoxenus/318.htm
>
> That link didn't work.

Oops, my bad. I left a few letters out.
http://www.ixpres.com/interval/monzo/aristoxenus/318tet.htm

> Ironically, Aristoxenus's scales are usually understood as
> subsets of 72-tET (q.v. Xenakis, etc.) You probably got
> 318-tET by observing the six circles of 12-tET in 72-tET
> and changing each of the six Pythagorean circles from 12-tET
> to 53-tET, yes?

I don't recall now exactly how I came up with 318-tET, but
I know for sure that it didn't have anything to do with 72-tET.
I wasn't examining 72-tET at all in relation to Aristoxenus,
at least not here in "My Conclusions". I was working strictly
with the diagram I drew there.

It *did* have a lot to do with 53-tET: I noticed that 53-tET
nicely fit the subdivisions of both of the Pythagorean semitones,
4 steps per limma and 5 steps per apotome.

Somehow as I worked on the diagram in "My Conclusions" I realized
that 318-tET gave extremely close approximations to the divisions
I calculated according to my translations of Aristoxenus's treatise.

My apologies if you go back and read parts of this webpage to
understand what I'm doing here at the end. I never finished it,
and in some places it's pretty confusing. Plus, last fall I
came up with a whole new interpretation of Aristoxenus's interval
measurements that I never added to this webpage at all.

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

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

> > Ironically, Aristoxenus's scales are usually understood as
> > subsets of 72-tET (q.v. Xenakis, etc.) You probably got
> > 318-tET by observing the six circles of 12-tET in 72-tET
> > and changing each of the six Pythagorean circles from 12-tET
> > to 53-tET, yes?
>
>
> I don't recall now exactly how I came up with 318-tET, but
> I know for sure that it didn't have anything to do with 72-tET.
> I wasn't examining 72-tET at all in relation to Aristoxenus,
> at least not here in "My Conclusions". I was working strictly
> with the diagram I drew there.

Well it does look like you saw Aristoxenus's scales as encompassing
six interlocking Pythagorean circles:

"every 6th tick-mark [notated 'xx'] designates 2^(1/53), the
smallest 'octave'-based temperament which accurately
represents the Pythagorean"

(from your page)

monz wrote:

> I don't recall now exactly how I came up with 318-tET, but
> I know for sure that it didn't have anything to do with 72-tET.
> I wasn't examining 72-tET at all in relation to Aristoxenus,
> at least not here in "My Conclusions". I was working strictly
> with the diagram I drew there.

You do say "144-tET - a useful approximation to Aristoxenus's system,
much less sophisticated than 318-tET" which kind of suggests you'd
compared the two (given you know that 72*2=144).

Have you considered "slack" in place of "relaxed"? It's shorter and
sounds more like "soft".

You say "... the 'octave', '4th', and '5th' are to be determined by
ear, which means that they will be measured as their usual Pythagorean
ratios 1:1, 4:3, and 3:2 ..." where 1:1 should be 2:1.

Graham

--- In tuning@y..., "Graham Breed" <graham@m...> wrote:

/tuning/topicId_20929.html#21582

> monz wrote:
>
> > I don't recall now exactly how I came up with 318-tET, but
> > I know for sure that it didn't have anything to do with 72-tET.
> > I wasn't examining 72-tET at all in relation to Aristoxenus,
> > at least not here in "My Conclusions". I was working strictly
> > with the diagram I drew there.
>
> You do say "144-tET - a useful approximation to Aristoxenus's
> system, much less sophisticated than 318-tET" which kind of
> suggests you'd compared the two (given you know that 72*2=144).

Yes, but that's in connection with older, more-established
interpretations of Aristoxenus's theory. 144-tET is the
"standard" view of Aristoxenus's divisions, stemming primarily
from Cleonides's redaction of Aristoxenus's theories [c. 100 AD],
and Westphal's attempted reconstruction of the actual treatise
[1883-93]. (Full citations are given at the end of my webpage.)

While working on my diagram and concentrating on the possible
Pythagorean aspects of Aristoxenus's theory, I wasn't thinking
at all of 72- or 144-EDO. In fact, that's specifically what
started my work on this webpage: I was writing a post to this
list to refute (or at least present some other possibilities)
Paul's mention of 72- and 144-EDO in connection with Aristoxenus,
and I got totally carried away.

> Have you considered "slack" in place of "relaxed"? It's shorter
> and sounds more like "soft".

Not a bad idea. I've gotten somewhat attached to "relaxed" at
this point, but it's worth consideration. Feedback welcome.
In any case, "soft" is totally inappropriate because it fails
to convey the essential basis of Aristoxenus's theory, with
its dichotomy between "tense" and "relaxed" in place of
Pythagorean-style string-length measurements.

> You say "... the 'octave', '4th', and '5th' are to be determined
> by ear, which means that they will be measured as their usual
> Pythagorean ratios 1:1, 4:3, and 3:2 ..." where 1:1 should be 2:1.

Oops! - my bad. Thanks Graham.

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

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21542

Looking at the table at the bottom of Rick Tagawa's
> homepage, http://sites.netscape.net/masanoritagawa/homepage, you'll
> see a few of these ambiguities:
>
> 13:11 "=" 20:17
> 13:10 "=" 22:17
>
> etc.
>

Thank you so much, Paul, for your assistance with 72-tET. There is
more to learn here... I can see that possibly it is the MOST useful
and PRACTICAL notation for Just Intonation... However, there's not a
lot on Just Intonational practice in 72-tET. I have the Maneri book
in front of me, and it doesn't seem to proceed from that approach.
Admittedly, I have not yet studied it thoroughly... (but will!)

I rather wish that Rick Tagawa would "clean up" this page. Isn't it
a little "messy"..? with e-mail addresses all over the place, etc.,
etc. They could at least have been put in brackets in a smaller
font... This is about the only Webpage on 72-tET that I know of!
Anyway...

A quote: (from an unidentified e-mail)

"Our friend Scott Van Duyne from CCRMA, Stanford has kindly provided
the following information: Preliminary Studies in the Virtual Pitch
Continuum by Joe Maneri and Scott Van Duyne (Published by Accentuate
Music, 42 Cornell Dr., Plainview, NY 11803) is an entire book on
composition in the 72 note scale. It runs in a sort of work book
style, but there are many short musical examples and a variety of
practical pitch organization approaches. It doesn't really deal with
just intonation. For that, try Ezra Sim's article in Computer Music
Journal from about 1988 or so, which arranges the 72 notes into
scales of 18 tones, and transposes around among them."

Well, this is nice... but finding a _Computer Music Journal_
from "about 1988 or so" with NO TITLE OR AUTHOR, might not be such an
easy task.... Does anybody have something a bit more SPECIFIC??

Maybe that will help with my understanding of 72-tET for Just
Intonation composition...

Daniel Wolf e-mail... pretty much summing up what was just stated
above:

"Joe Manieri teaches 72-tone equal temperament at New England
Conservatory without any regard to its excellent approximations of
simple-integer ratios. This, essentially, is the same approach taken
by Carrillo, Haba and Wyschnegradsky in an earlier generation or by
Ferneyhough & Co. nowadays: the use of microtones to create a highly
inflected musical surface. One of Partch's initial impulses, to
accurately represent the inflections of a speaking voice was not
unrelated to this view. The alternative approach, whose pioneers were
Lourie, Novaro, Partch (slightly later), Fokker, and Wilson, is
probably more central to the interests of the tuning list membership,
is certainly a deeper view of musical materials, but one which does
not automatically dismiss the other aesthetic."

An e-mail from James McCartney....Anybody know who HE is?? He must
have been on the list some time ago (??):

"I have been using 72 equal quite a bit. One thing that has helped me
immensely is a 72TET 'Bingo' card I made up.

58 9 32 55 56 29 52 3 26
16 39 62 13 36 59 10 33 56
46 69 20 43 66 17 40 63 14
4 27 50 1 24 47 70 21 44
34 57 8 31 54 5 28 51 2
64 15 38 61 12 35 58 9 32
22 45 68 19 42 65 16 39 62
52 3 26 49 0 23 46 69 20
10 33 56 7 30 53 4 27 50
40 63 14 37 60 11 34 57 8
70 21 44 67 18 41 64 15 38
28 51 2 25 48 71 22 45 68
58 9 32 55 56 29 52 3 26
16 39 62 13 36 59 10 33 56
46 69 20 43 66 17 40 63 14

"~3/2's go up and ~5/4's go right. On the card all intervals are
uniformly spaced, that is a 7/6 for example is two right and one up."

Huh? How is he getting this??

If I go 5/4 * 5/4 = 25/16
25/16 * 3/2 = 75/32...

That's not 7/6

Well, maybe I should have DIVIDED by the 3/2. I'll try it again:

5/4 * 5/4 = 25/16
25/16 * 2/3 = 25/24...

That's STILLl not 7/6!

What am I doing wrong here??

"Some interval equivalences: 3/2 = 42, 9/8 = 12, 4/3 = 30, 16/9 = 60,
15/8 = 65, 5/4 = 23, 5/3 = 53, 10/9 = 11, 9/5 = 61, 6/5 = 19, 8/5 =
49, 16/15 = 7, 7/5 = 35, 7/4 = 58, 7/6 = 16, 14/9 = 46, 11/8 = 33..
etc. Many interesting symmetries and enharmonicities can be found.. --
- james mccartney

Anyway, a table like this, if I could get it to work correctly, could
be quite helpful.

Now Paul, the following was from your OWN post. Recently, both Joe
Monzo and Dan Sterns have been advocating 144-tET as being able to
approximate almost ANY tuning system. However, you make the
following statement:

[Paul Erlich]:

"If 13- or higher-limit intervals were included, using 144-tET to
approximate them could lead to dangerous inconsistencies, since 144-
tET is only consistent through the 11-limit."

I believe, Paul, that a long time ago we went over this and I've
forgotten it, but it seems mighty peculiar that an ET with TWICE the
number of steps would be LESS consistent than one with HALF the
number...

Could you please briefly go over that again??

Regarding your description:

> Once you hit the 19-limit, 72-tET is inconsistent. For example, the
> best approximation of 8:19 is 90 steps of 72-tET; the best
> approximation of 13:19 is 39 steps of 72-tET; but if you use these
> approximations to construct an 8:13:19 triad, 8:13 would have to be
> approximated by 90 - 39 = 51 steps of 72-tET -- but we've seen that
> 50 steps is a better approximation.
> Again, the stuff at the bottom of Rick's homepage should elaborate
on this and make it clearer . . .

Actually, do you realize that the paragraph you wrote above is MUCH
CLEARER than the similar description on the Tagawa page?!

I got it right away in the recent post.... I think the "subtraction"
is what did it!

_________ ______ _______
Joseph Pehrson

Hi Paul,

What do you think about 612 as a 51 gear of 12-teT.

And 3125 as a 5 gear of 5 gear of 5 gear of 5 gear of 5-teT :-)

Kees

----- Original Message -----
From: "Paul H. Erlich" <PERLICH@ACADIAN-ASSET.COM>
To: <tuning@yahoogroups.com>
Sent: Tuesday, April 24, 2001 11:29 PM
Subject: [tuning] Re: Reply to Joseph Pehrson on 72-tET

> Hi Joseph!
>
> The idea of 72-tET as a six-gear version of a 12-tET bike reminds me of
> something similar we both learned when talking about your 19-tone just
> scales:
>
> 171-tET (13-limit-consistent, 9-limit-ULTRA-accurate) as a nine-gear
version
> of 19-tET. Do you remember that? The difference is that 19-tET
approximates
> a chain of just minor thirds, not fifths, and that's kind of a weird
lattice
> direction for the bike gears to point in, if you know what I mean.
>
> I found some others "bikes" among the super-accurate ETs (probably of
> theoretical interest only and completely useless):
>
> 217-tET (19-limit-consistent, 13-limit-very-accurate) as a seven-gear
> version of 31-tET. Here 31-tET approximates a chain of just major thirds
or
> just harmonic sevenths or septimal tritones. Three more weird gear
> directions.
>
> 342-tET (11-limit-ULTRA-accurate) as a two-gear version of 171-tET, itself
a
> nine-gear version of 19-tET. Here's a bike you can really cruise the
lattice
> on, hugging those minor third grooves, of course.
>
> 494-tET (15-limit-ULTRA-accurate, 17-limit consistent) as a 26-gear
version
> of 19-tET, OR as a 19-gear version of 26-tET (which can be thought of as a
> chain of just harmonic sevenths)
>
> 665-tET (3-limit-HYPER-accurate, 9-limit consistent) as a 35-gear version
of
> 19-tET.
>
> 1178-tET (13-limit-ULTRA-accurate, 21-limit consistent) as a 62-gear
version
> of 19-tET.
>
> 1330-tET (3-limit-HYPER-accurate, 11-limit consistent) as a 2-gear version
> of 665-tET, itself a 35-gear version of 19-tET.
>
>
> Very accurate > 99.7% accuracy
> ULTRA-accurate > 99.987% accuracy
> HYPER-accurate > 99.9999999978% accuracy
>
> You do not need web access to participate. You may subscribe through
> email. Send an empty email to one of these addresses:
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>
>
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>
>
>

--- In tuning@y..., "Paul H. Erlich" <PERLICH@A...> wrote:

/tuning/topicId_20929.html#21567

> Hi Joseph!
>
> The idea of 72-tET as a six-gear version of a 12-tET bike reminds
me of something similar we both learned when talking about your 19-
tone just scales:
>
> 171-tET (13-limit-consistent, 9-limit-ULTRA-accurate) as a nine-
gear version of 19-tET. Do you remember that? The difference is that
19-tET approximates a chain of just minor thirds, not fifths, and
that's kind of a weird lattice direction for the bike gears to point
in, if you know what I mean.
>

Hi Paul!

I'm glad you enjoyed my "bike" metaphor! Well, it's definitely
something that came to mind when I thought about these 12-tET
chains! Unfortunately, I can't remember the 19-tET situation, but
thanks for re-posting it, as well as all the other instances
of "bikes."

We're really turning out to be "bikers" on this list!

_________ _____ _____ ____
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:
>
> Thank you so much, Paul, for your assistance with 72-tET. There is
> more to learn here... I can see that possibly it is the MOST useful
> and PRACTICAL notation for Just Intonation... However, there's not
a
> lot on Just Intonational practice in 72-tET. I have the Maneri
book
> in front of me, and it doesn't seem to proceed from that approach.
> Admittedly, I have not yet studied it thoroughly... (but will!)

No you're right, Maneri & Co. don't mention JI, don't care about JI.
To them, dissonance has already been emancipated, so there's no point
trying to classify consonance and dissonance anymore.
> >

> An e-mail from James McCartney....Anybody know who HE is??

I saw a guitar in a garbage can once with his name on it (no
kidding!). But no, I don't know who he is.

>
>
> "I have been using 72 equal quite a bit. One thing that has helped
me
> immensely is a 72TET 'Bingo' card I made up.
>
> 58 9 32 55 56 29 52 3 26
> 16 39 62 13 36 59 10 33 56
> 46 69 20 43 66 17 40 63 14
> 4 27 50 1 24 47 70 21 44
> 34 57 8 31 54 5 28 51 2
> 64 15 38 61 12 35 58 9 32
> 22 45 68 19 42 65 16 39 62
> 52 3 26 49 0 23 46 69 20
> 10 33 56 7 30 53 4 27 50
> 40 63 14 37 60 11 34 57 8
> 70 21 44 67 18 41 64 15 38
> 28 51 2 25 48 71 22 45 68
> 58 9 32 55 56 29 52 3 26
> 16 39 62 13 36 59 10 33 56
> 46 69 20 43 66 17 40 63 14
>
>
> "~3/2's go up and ~5/4's go right. On the card all intervals are
> uniformly spaced, that is a 7/6 for example is two right and one
up."
>
> Huh? How is he getting this??
>
> If I go 5/4 * 5/4 = 25/16
> 25/16 * 3/2 = 75/32...
>
> That's not 7/6

Ah, but in 72-tET, it is! 75/64, that is, is approximated by the same
interval as 7/6. James's point is that no matter where you start on
the bingo card, if you go two right and one up, you add 16 (mod 72)
to the number you started with -- which corresponds to moving up an
approximate 7:6 in pitch. This is a sensible way of thinking about it
because 7:6 is something you can immediately hear (if you've
completed your 11-limit ear training), while 75:64 only makes sense
if you combine several 5-limit intervals end to end.
>
>
>
> Now Paul, the following was from your OWN post. Recently, both Joe
> Monzo and Dan Sterns have been advocating 144-tET as being able to
> approximate almost ANY tuning system. However, you make the
> following statement:
>
> [Paul Erlich]:
>
> "If 13- or higher-limit intervals were included, using 144-tET to
> approximate them could lead to dangerous inconsistencies, since 144-
> tET is only consistent through the 11-limit."
>
> I believe, Paul, that a long time ago we went over this and I've
> forgotten it, but it seems mighty peculiar that an ET with TWICE
the
> number of steps would be LESS consistent than one with HALF the
> number...
>
> Could you please briefly go over that again??

Joseph, you're familiar with the fact that 24-tET is inconsistent in
the 7-limit, right? But 12-tET is consistent in the 7-limit (just not
terribly accurate). So there again, an ET with TWICE the number of
steps is less consistent than one with half the number. I would
definitely prefer to notate 7-limit JI in 12-tET than in 24-tET. Does
that clear it up?

BTW, different people have different opinions about how high in the
ETs does consistency really matter. As far as I'm concerned, in 35-
tET and higher ETs, it kind of doesn't matter, you can just pick your
approximations and run with them, even if they're not the best
approximations the ET has to offer. For example, in 64-tET there are
at least three tunings for a major triad that are still recognizable
major triads!

> Actually, do you realize that the paragraph you wrote above is MUCH
> CLEARER than the similar description on the Tagawa page?!
>
> I got it right away in the recent post.... I think
the "subtraction"
> is what did it!

I'm glad I'm improving!

--- In tuning@y..., "Kees van Prooijen" <kees@d...> wrote:
> Hi Paul,
>
> What do you think about 612 as a 51 gear of 12-tET.

I don't -- none of the 12-tET intervals correspond to 612's best
approximation to any consonant JI interval.
>
> And 3125 as a 5 gear of 5 gear of 5 gear of 5 gear of 5-teT :-)
>
I did notice 3125-tET but it also didn't seem to fit the analogy,
just as 612 didn't.

--- In tuning@y..., jpehrson@r... wrote:

> Unfortunately, I can't remember the 19-tET situation

Remember how you were finding that, when you compared various 7-limit
JI scales to 19-tET, you kept seeing cent deviations that were
multiples of 7 cents? We resolved this by noting that 7 cents is one
step of 171-tET, which is nine 19-tET chains, and that 171-tET is a
very accurate approximation of 7-limit JI. Remember this?

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21605>

No you're right, Maneri & Co. don't mention JI, don't care about JI.
> To them, dissonance has already been emancipated, so there's no
point trying to classify consonance and dissonance anymore.
> > >

Got it! I should have guessed at this esthetic from his performances
at Johnny Reinhard's microthon...
> >

> > Huh? How is he getting this??
> >
> > If I go 5/4 * 5/4 = 25/16
> > 25/16 * 3/2 = 75/32...
> >
> > That's not 7/6
>

> Ah, but in 72-tET, it is! 75/64, that is, is approximated by the
same interval as 7/6.

Am not sure I'm getting the point, then, in this. Is he comparing 72-
tET to a 5-limit system??

> Joseph, you're familiar with the fact that 24-tET is inconsistent
in the 7-limit, right? But 12-tET is consistent in the 7-limit (just
not terribly accurate). So there again, an ET with TWICE the number
of steps is less consistent than one with half the number. I would
> definitely prefer to notate 7-limit JI in 12-tET than in 24-tET.
Does that clear it up?
>

Oh! I guess I was just a bit confused again about "consistency..."
But, if I'm understanding this correctly, it would make sense that a
scale with a greater number of units could possible be less
"consistent" for the same chord... there would more notes that could
approximate it... (??)

________ _______ ______
Joseph Pehrson

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21608

> --- In tuning@y..., jpehrson@r... wrote:
>
> > Unfortunately, I can't remember the 19-tET situation
>
> Remember how you were finding that, when you compared various 7-
limit JI scales to 19-tET, you kept seeing cent deviations that were
> multiples of 7 cents? We resolved this by noting that 7 cents is
one step of 171-tET, which is nine 19-tET chains, and that 171-tET
is a very accurate approximation of 7-limit JI. Remember this?

Oh of course! I remember this now! This was the reason that every 7-
limit JI pitch deviated from 19-tET by some multiple of 7 cents,
depending on which other "bike chain" it fell....

________ ______ _____ _
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

> > Ah, but in 72-tET, it is! 75/64, that is, is approximated by the
> same interval as 7/6.
>
> Am not sure I'm getting the point, then, in this. Is he comparing
72-
> tET to a 5-limit system??

Well, you can map out the entire 72-tET system as a 5-limit system,
which is what he's doing . . . but that doesn't preclude you from
observing and using 7- and higher-limit relationships.

--- In tuning@y..., jpehrson@r... wrote:
>
> Oh of course! I remember this now! This was the reason that every
7-
> limit JI pitch deviated from 19-tET by some multiple of 7 cents,
> depending on which other "bike chain" it fell....

Exactly . . . this was the 171-tET bike "in action"!

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21646

> --- In tuning@y..., jpehrson@r... wrote:
>
> > > Ah, but in 72-tET, it is! 75/64, that is, is approximated
> > > by the same interval as 7/6.
> >
> > Am not sure I'm getting the point, then, in this. Is he
> > comparing 72-tET to a 5-limit system??
>
> Well, you can map out the entire 72-tET system as a 5-limit
> system, which is what he's doing . . . but that doesn't
> preclude you from observing and using 7- and higher-limit
> relationships.

Joe, what Paul is saying here is that in Rick's diagram the
small interval separating 75:64 from 7:6 vanishes. This
interval is the same ol' 225:224 "kleisma" about which I
wrote a few days ago in connection with the audible similarity
of 225:128 and 7:4. The kleisma is about 7.7 cents.

It was used by Fokker in his 1949 book _Just Intonation_ to
teach singers how to sing 7-limit intervals in tune, by relating
the 7-limit ratios to their kleismatic near-equivalent 5-limit
cousins, with which the singers would already be familiar.
Using "C" as 1/1, Fokker demonstrates how the 225:128 "A#"
sounds very much like the 7:4 "Bb". (I posted a summary of
this book in 1999.)

However, a while back I posted some interesting observations
about this interval.

In retuning my _3 Plus 4_ (which is in D#-minor / F#-major)
into JI, I found that at the climax of the section before the
"hook", the D#-minor chord only sounded "right" to me with
a minor third of the very 75:64 [= ~274.582 cents] which Rick
confounds with 7:6 [= ~266.87 cents].

But interestingly, I had tried 7:6 first and it didn't work.

I also tried 19:16 [= ~297.513 cents] for the "minor 3rd",
and liked it much better than 7:6 but not as much as 75:64.

The point? 19:16 is *much* farther away from 75:64 [~22.931
cents, a kind of "comma"] than 7:6 is [the kleisma, ~7.7 cents],
and yet the first two were the ones that had the similar "affect"
to my ears. I was quite surprised about this.

So while in *most* cases it's perfectly OK to not bother
distinguishing between 7:6 and 75:64, in my particular case
it was very important to do so. This is one of the fascinating
things I find about working in JI - that there are such subtle
affects that creep into your music when you're *not* ignoring
tiny differences in intonation.

There was quite a bit of discussion on this after I posted it.
(Search the archives - all this stuff is in there.)

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

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21647

> --- In tuning@y..., jpehrson@r... wrote:
> >
> > Oh of course! I remember this now! This was the reason
> > that every 7-limit JI pitch deviated from 19-tET by some
> > multiple of 7 cents, depending on which other "bike chain"
> > it fell....
>
> Exactly . . . this was the 171-tET bike "in action"!

Just thought I'd point out that the prolific German tuning
theorist Martin Vogel is a big advocate of 171-tET. He even
had a keyboard built in this tuning. The info is available
in English in _On the Relations of Tone_, and in German in
several of his many books.

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

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> So while in *most* cases it's perfectly OK to not bother
> distinguishing between 7:6 and 75:64, in my particular case
> it was very important to do so. This is one of the fascinating
> things I find about working in JI - that there are such subtle
> affects that creep into your music when you're *not* ignoring
> tiny differences in intonation.
>
> There was quite a bit of discussion on this after I posted it.
> (Search the archives - all this stuff is in there.)
>
To summarize, my reaction was that I found it highly unlikely that
your ear was really targeting a 75:64 _harmonic interval_ -- 75:64
can be built from two 5:4s and one 3:2 but there was no such
construction operating in your piece -- rather it seemed that you
simply wanted a small minor third of a particular width.

I should emphasize that if you did go up two 5:4s and one 3:2 in 72-
tET, you'd end up with an interval which, in JI terms, "should" be a
75:64 but happens to be closer to 7:6. You mentioned that the
difference between the two is about 7 cents (yes, 1 degree of 171-
tET). But no 7-cent errors actually afflict any tunable intervals
here. Each 3:2 is 2 cents flat, and the 5:4 is 2 cents flat, so the
end result of proceeding by these intervals is 6 cents flatter that
in would be in JI. But even if you can hear a 6-cent difference, you
probably won't care, unless you're _so_ familiar with the experience
of complex progressions in strict JI that the sound of the end result
of this chain is etched in your mind's ear.

Now the fact that the end result of this chain in 72-tET happens to
be within 1 cent of a just 7:6 is something you can exploit
compositionally (as James Tenney has done). You can, for example,
sound the beginning and the end of the chain together and it'll sound
like a wonderfully consonant 7:6.

This is analogous to the I-vi-ii-V progression in a meantone
temperament (NOT 72-tET). If you played this chain of intervals in
JI, you'd end up with a V that was 40:27 from the I. 40:27 is 21.5
cents (a comma) different from 3:2. But in meantone the end result of
this chain of intervals is much closer to 3:2. This allows you to
have a V which is consonant with the I, forming a near-3:2 ratio. I
think most people would be pleased, rather than upset, that the JI
behavior was thwarted here -- and yet no consonant interval was off
by more than 1/4 comma, or ~5 cents -- so no 21.5-cent deviation from
JI is evident to the ear (unless, again, you _so_ well-trained in
strict JI (or 72-tET) that the sound of the 40:27 is etched in your
mind).

--- In tuning@y..., PERLICH@A... wrote:

> To summarize, my reaction was that I found it highly unlikely that
> your ear was really targeting a 75:64 _harmonic interval_ -- 75:64
> can be built from two 5:4s and one 3:2 but there was no such
> construction operating in your piece -- rather it seemed that you
> simply wanted a small minor third of a particular width.

Right you are, Paul. I recall that I wrote about this too in
the old posts. When 7:6 didn't work, which surprised me because
I thought that was the sound I was going for, I simply manipulated
the amount of pitch-bend on the note until it was what I wanted.
IIRC, it turned out to be ~279 cents, which is only ~4.5 cents
higher than 75:64.

Since my intention was to retune _3 Plus 4_ into JI - a flavor
of JI that was essentially 5-limit with occasionally sprinklings
of 7- and 11-limit ratios for added spice - I settled on tuning
and notating this pitch as 75:64. You are correct that I didn't
reach it thru any particular audible combination of 3:2s and
5:4s.

This alteration, from the ~279 pitch I reached by ear to 75:64,
didn't change the affect I wanted and was hearing. What surprised
me was that there was such a noticeable difference between 7:6
and 75:64.

> ... But even if you can hear a 6-cent difference, you
> probably won't care, unless you're _so_ familiar with the experience
> of complex progressions in strict JI that the sound of the end result
> of this chain is etched in your mind's ear.

Excellent point.

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

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

/tuning/topicId_20929.html#21695

>
> Joe, what Paul is saying here is that in Rick's diagram the
> small interval separating 75:64 from 7:6 vanishes. This
> interval is the same ol' 225:224 "kleisma" about which I
> wrote a few days ago in connection with the audible similarity
> of 225:128 and 7:4. The kleisma is about 7.7 cents.
>

Hi Monz!

Ok... FINALLY I get the arithmetic on this. I was getting a 75/32
and that would be the same as a 75/64 in a different octave, yes??

I believe the chart was actually by James McCartney, and not by Rick
Tagawa...

It's interesting, though, since it's a kind of 5-limit
"interpretation" of 72-tET... but Paul's chart of the just intervals
on that website is even MORE useful and interesting!

> However, a while back I posted some interesting observations
> about this interval.
>
> In retuning my _3 Plus 4_ (which is in D#-minor / F#-major)
> into JI, I found that at the climax of the section before the
> "hook", the D#-minor chord only sounded "right" to me with
> a minor third of the very 75:64 [= ~274.582 cents] which Rick
> confounds with 7:6 [= ~266.87 cents].
>
> So while in *most* cases it's perfectly OK to not bother
> distinguishing between 7:6 and 75:64, in my particular case
> it was very important to do so. This is one of the fascinating
> things I find about working in JI - that there are such subtle
> affects that creep into your music when you're *not* ignoring
> tiny differences in intonation.

This is really interesting... its an entirely different "level" of
perception, so it seems....
_______ _______ ____ _
Joseph Pehrson

_________ ________ _______ _____
Joseph Pehrson

--- In tuning@y..., PERLICH@A... wrote:

/tuning/topicId_20929.html#21702

> --- In tuning@y..., "monz" <joemonz@y...> wrote:
> >
> > So while in *most* cases it's perfectly OK to not bother
> > distinguishing between 7:6 and 75:64, in my particular case
> > it was very important to do so. This is one of the fascinating
> > things I find about working in JI - that there are such subtle
> > affects that creep into your music when you're *not* ignoring
> > tiny differences in intonation.
> >
> > There was quite a bit of discussion on this after I posted it.
> > (Search the archives - all this stuff is in there.)
> >
> To summarize, my reaction was that I found it highly unlikely that
> your ear was really targeting a 75:64 _harmonic interval_ -- 75:64
> can be built from two 5:4s and one 3:2 but there was no such
> construction operating in your piece -- rather it seemed that you
> simply wanted a small minor third of a particular width.
>
> I should emphasize that if you did go up two 5:4s and one 3:2 in 72-
> tET, you'd end up with an interval which, in JI terms, "should" be
a
> 75:64 but happens to be closer to 7:6. You mentioned that the
> difference between the two is about 7 cents (yes, 1 degree of 171-
> tET). But no 7-cent errors actually afflict any tunable intervals
> here. Each 3:2 is 2 cents flat, and the 5:4 is 2 cents flat, so the
> end result of proceeding by these intervals is 6 cents flatter that
> in would be in JI. But even if you can hear a 6-cent difference,
you probably won't care, unless you're _so_ familiar with the
experience of complex progressions in strict JI that the sound of
the end result of this chain is etched in your mind's ear.

OK... so wassap here...??

I've heard Monzo's 3+4. It's one of my _favorite_ xenharmonic
pieces! And yes, it *does* seem that the use of the particular third
matters...

But Paul Erlich seems to be implying, Monz, that you really couldn't
be hearing the difference between 7:6 and 75:64 in your piece.

Is that true??

_________ ____ ___ ____
Joseph Pehrson

--- In tuning@y..., jpehrson@r... wrote:

> But Paul Erlich seems to be implying, Monz, that you really couldn't
> be hearing the difference between 7:6 and 75:64 in your piece.
>
> Is that true??

No, that's not what I was implying. Monz clearly understood what I was implying, as his reply
makes clear. In fact Monz seems to be admitting (perhaps for the first time) that he actually
targeted an interval he heard by ear, and then altered that pitch slightly so that it could be
expressed as a 5-prime-limit ratio, for no other reason than wanting the piece to be in
5-prime-limit RI. The interval he heard by ear was clearly larger than 7:6 . . . and if he had to use
72-tET, he might have used the 1/12-tone flat minor third to represent it . . . even though 7:6 and
75:64 are both represented by a 1/6-tone flat minor third in 72-tET.

The real point of my message that should be of interest to you, in considering 72-tET, was
well-illustrated in the meantone I-vi-ii-V analogy I was making there . . . if you're still interested,
feel free to go back and read that analogy and tell me if it makes sense to you.

--- In tuning@y..., paul@s... wrote:

/tuning/topicId_20929.html#21768

> The real point of my message that should be of interest to you, in
considering 72-tET, was well-illustrated in the meantone I-vi-ii-V
analogy I was making there . . . if you're still interested, feel
free to go back and read that analogy and tell me if it makes sense
to you.

Thanks, Paul... I believe I posted that that "reprise" was very
valuable for me, and I saved it... I can see how eliminating the
comma can make for a greater multiplicity of consonant intervals in
ANY of the systems...

_________ ______ _____ _
Joseph Pehrson