Yahoo Tuning Groups Ultimate Backup tuning JI notation

In-Reply-To: <8u6343+g943@eGroups.com>
Monzo wrote:

> Even tho I love Johnston's music and must acknowledge that
> his theories are very similar to my own, I want to emphasize,
> in agreement with Daniel Wolf, that Johnston's notation,
> while admirably compact, is needlessly complicated. It forces
> the reader to understand the 5-limit lattice _a priori_ and
> to calculate all other prime-factor adjustments thru a 2-step
> process in relation to the 5-limit lattice.

Hey, it's another notation holy war! Yippeee!!

> Wolf and I both propose modifications of Johnston's ideas
> which utilize the linear 3-limit (Pythagorean) tuning as
> the _a priori_ basis, since 'standard' notation developed
> on that basis anyway: each distinctive letter-name and all
> of the associated (and multiple) sharps and flats by themselves
> clearly indicate an extended Pythagorean system. Use of prime-
> factor 5 is indicated by further accidentals (which are also
> required by Johnston anyway), and the other primes follow in
> a similar manner.

I've devised a subtly different system, which combines certain aspects of
both the others. Some clues at http://x31eq.com/schismic.htm .
It also has a linear 3-limit-ish basis, but places the wolf in an
unexpected place, so the white notes are (or could be) almost the same
places as in Johnston's notation.

The idea is to take 12 notes, fix them, and write them as you would
anyway. In addition, the # and b symbols have consistent meanings, so you
could get by with only remembering 7 notes plus the sharpness. Then, add
a "comma shift" symbol, and use schismic approximations.

> These systems have the big advantage of requiring _a priori_
> systemic knowledge only of the linear Pythagorean system: any
> pitch can be calculated by its relation to the Pythagorean tuning,
> which in turn can also be calculated from 1/1 by knowledge of the
> simple 'rules' of the generating cycle of 4ths/5ths and the
> letter-names and sharps/flats.

Your system requires:

tonic pitch 1
size of fifth 1
four modifiers 4

That makes 6 free variables. Mine requires

7 pitches 7
2 modifiers 2

which is 9 free parameters. So 50% more complex. Although it could also
be

tonic pitch 1
size of fifth 1
one modifier 1

which is 3 free parameters, so only half as complex as yours. It all
depends on what we expect the performer to work out. I expect they'll
remember those 7 pitches either way. They could even be built in to the
instrument somehow.

The advantages of my system are that there's only one new modifier to
remember, and you don't have to remember whether G# is supposed to be
higher or lower than Ab. Also, a diatonic scale on the white notes can be
written with the minimum number of modifiers (I expect the key signature
will usually override this default).

It also means that it isn't too drastic if the performer assumes the
original scale is 12-equal, and has a good ear for 5-limit chords, because
it's sufficiently vague to start with. For performers without a good ear,
there will be a problem with precision -- you can only nail the 11-limit
to within about 10 cents. This improves with lower limits: 5-limit can be
considered JI.

72-equal notation could work in a similar way. Different symbols for
commas and quartertones would be useful. That's all been discussed before
as well.

> Henry, you might want to try to talk this composer into letting
> you copy his score using one of these notation systems instead.
> In my opinion, it would ultimately encourage more performances,
> even tho at the present time I must admit that certainly more
> performers are familiar with Johnston's notation than with mine
> or Wolf's.

Well, nobody will be at all familiar with my system. In fact, nobody
seems to like it. Even I prefer meantone notation, when I actually write
anything down. It's about the same in the 11-limit, anyway.

> Perhaps the only way to make a microtonal notation more 'natural'
> is to show its pitches's adjustments in relation to 12-tET,
> because that's the scale with which everyone is familiar,
> and because the accursed hardware and software designers insist
> on building it into every instrument and program they create.
>
> But it's very difficult to indicate JI tunings *systematically*
> in relation to 12-tET, because of the inherent differences
> between ETs (which are closed systems) and JI (which are open,
> i.e., potentially infinite).

That's where the 72= notation would come in handy. It's 11-limit
consistent, and can be replaced with cents deviations, even if they won't
be accurate to the JI. As has been discussed before.

Graham

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

>> But it's very difficult to indicate JI tunings *systematically*
>> in relation to 12-tET, because of the inherent differences
>> between ETs (which are closed systems) and JI (which are open,
>> i.e., potentially infinite).

>That's where the 72= notation would come in handy. It's 11-limit
>consistent, and can be replaced with cents deviations, even if they won't
>be accurate to the JI. As has been discussed before.

I agree with that completely, too, though 72= is 17-limit consistent (it's
_unique_ through the 11-limit).
But the point is, for live musicians in today's world, there's no way you're
going to get more accuracy by assuming Pythagorean as the default than by
assuming 12= as the default. Assuming fine, conservatory-trained musicians,
learning to interpolate 5 new positions within the already-familiar
semitones (using 11-limit JI intervals as a guide, reliable to within 4¢) is
far more doable than learning a whole new set of unaltered pitches and then
a whole new set of alterations applicable to any of unaltered pitches. As to
the latter, I would basically say, don't even try.

🔗Joseph Pehrson <josephpehrson@compuserve.com>

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
> >> But it's very difficult to indicate JI tunings *systematically*
> >> in relation to 12-tET, because of the inherent differences
> >> between ETs (which are closed systems) and JI (which are open,
> >> i.e., potentially infinite).
>
> >That's where the 72= notation would come in handy. It's 11-limit
> >consistent, and can be replaced with cents deviations, even if
they won't be accurate to the JI. As has been discussed before.
>
> I agree with that completely, too, though 72= is 17-limit
consistent (it's_unique_ through the 11-limit).

> But the point is, for live musicians in today's world, there's no
way you're going to get more accuracy by assuming Pythagorean as the
default than by assuming 12= as the default. Assuming fine,
conservatory-trained musicians, learning to interpolate 5 new
positions within the already-familiar semitones (using 11-limit JI
intervals as a guide, reliable to within 4¢) is far more doable
than learning a whole new set of unaltered pitches
and then a whole new set of alterations applicable to any of
unaltered pitches. As to the latter, I would basically say, don't
even
try.

Ummm. This stuff is important... so I want to make sure I'm
understanding it!

Basically Wolf and Monz want to set up a system that is alterable
from 3-limit Pythagorean. Essentially this is pretty close, anyway,
to 12-tET, since 12-tET is fundamentally derived from it... so Paul
suggests why not "go with the flow" and actually use 12-tET as the
basis. (Although it could slightly offend JI purists!)

This is, in fact, what the 72-tETters do... using the basic 12-tET
chromatic scale and adding 5 more pitches, equalling 72... They seem
to be quite a practical bunch all in all...

HOWEVER, it seems as though Ben Johnston's system would require a
DIFFERENT set of basic pitches than Pythagorean OR 12-tET as the
basis. HIS basis is 5-limit and he makes alterations from there.

But then what is his chromatic scale?? Where does it come from,
piling up just thirds?? Must be (??) So HIS system is more related
to the "classic" method of creating just scales (??) How many
question marks can I write (???)

Thanks for the help! I don't have time right at this very moment to
research the Johnston book... unless there is a Website on it, so any
help would be apprciated!

Thanks!!

Joey

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

Joseph Pehrson wrote,

>But then what is [his] chromatic scale?? Where does it come from,
>piling up just thirds?? Must be (??) So HIS system is more related
>to the "classic" method of creating just scales (??) How many
>question marks can I write (???)

Ben Johnston starts with a _diatonic_ scale (the chromatic modifiers # and b
indicate fixed alterations of 25:24). The tuning of this diatonic scale is

C 1/1
D 9/8
E 5/4
F 4/3
G 3/2
A 5/3
B 15/8
C 2/1

His reasoning is that this is the "just C major scale" and what the ear
naturally wants. Unfortunately, this is untenable. Because of the dissonant
interval between D and A, at no time in the history of notated Western music
was this how the C major scale was tuned. The D and A would be a disaster
for string instruments such as the violin where the open D and A strings are
tuned to one another. Moreover, once the key moves away from C major, the
notational complications become monstrous.

🔗Joseph Pehrson <josephpehrson@compuserve.com>

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

http://www.egroups.com/message/tuning/15278

>
> C 1/1
> D 9/8
> E 5/4
> F 4/3
> G 3/2
> A 5/3
> B 15/8
> C 2/1
>
> His reasoning is that this is the "just C major scale" and what the
ear naturally wants. Unfortunately, this is untenable.

Well, isn't this funny... I just happened to be going through the new
version of Scala (1.7) and Manuel has placed this scale in the root
directory as the main one there!

Intense Diatonic Systonon, also Zarlino's scale
0: 1/1 0.000 unison, perfect prime
1: 9/8 203.910 major whole tone
2: 5/4 386.314 major third
3: 4/3 498.045 perfect fourth
4: 3/2 701.955 perfect fifth
5: 5/3 884.359 major sixth, BP sixth
6: 15/8 1088.269 classic major seventh
7: 2/1 1200.000 octave

I was just wondering about it. With all those just ratios it is,
indeed, "intense." But when did Zarlino come up with this?...
obviously not just yesterday... And isn't it peculiar that the
Bohlen-Pierce scale shares the 5/3 sixth... or is that just the way
the math works out...(??)

P.S. And what does "systonon" mean (??)

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Joseph Pehrson <josephpehrson@compuserve.com>

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

http://www.egroups.com/message/tuning/15280

Thanks, Paul, for the VERY quick answers to these questions! I get
the answer practically before I type the question!!

> Looks like a misspelling of "Syntoton", which is what Ptolemy
called scales like this -- and where the term "syntonic comma" came
from.

Oh, sure... I wouldn't have been confused with *that* spelling...

This scale was called by Ptolemy the "Intense Syntoton Diatonic" --
he described dozens of other scales and did not single this one out
as "special" in any way.
>
> I'm not sure why it's called "Zarlino's scale", since Zarlino lived
before the major scale superceded the modes of Renaissance polyphony,
and Zarlino himself proposed 2/7-comma meantone. Perhaps Margo or
someone could enlighten.
>

I THOUGHT we'd been talking about that...I hadn't identified him with
the "correction" of the syntonic comma in any way...

> Typically, this scale is just called "the JI major scale".
>

Ummm. Yep. This is one I should know (!) :)

> >And isn't it peculiar that the
> >Bohlen-Pierce scale shares the 5/3 sixth... or is that just the
way the math works out...(??)
>

> Joseph, 5:3 is the consonant "major sixth", and the BP scale
approximates it well since the ratio contains only odd numbers.

Oh... sure. The odd numbers are the "giveaway..."

thanks!!!

Joseph

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Monz <MONZ@JUNO.COM>

--- In tuning@egroups.com, "Joseph Pehrson" wrote:

> http://www.egroups.com/message/tuning/15279
>
>
> Intense Diatonic Systonon, also Zarlino's scale
> 0: 1/1 0.000 unison, perfect prime
> 1: 9/8 203.910 major whole tone
> 2: 5/4 386.314 major third
> 3: 4/3 498.045 perfect fourth
> 4: 3/2 701.955 perfect fifth
> 5: 5/3 884.359 major sixth, BP sixth
> 6: 15/8 1088.269 classic major seventh
> 7: 2/1 1200.000 octave
>
>
> I was just wondering about it. With all those just ratios it is,
> indeed, "intense." But when did Zarlino come up with this?...
> obviously not just yesterday...

This scale was first recorded by Ptolemy (100s AD, Alexandria).
Ptolemy's theories were rendered into Latin by Boethius c. 505,
and the scale was advocated by Zarlino in the mid-1500s, writing
in Italian, as the basis of musical harmony. Keep in mind that
Zarlino was also the first theorist to extend consideration of
harmonic units from dyads to triads; all other harmonic theoretical
speculation before him dealt with *intervals* (dyads).

As far as I know, the introduction of this tuning into the
German literature occurred in the early 1600s by Lippius and
then soon after by Rene Descartes (the philosopher).

> P.S. And what does "systonon" mean (??)

That's supposed to be 'syntonon'. It's Greek for 'tense'.

Use of the term originated with the theories of Aristoxenus.
See my paper (not finished and in need of much editing...sorry)
at:

http://www.ixpres.com/interval/monzo/aristoxenus/318tet.htm

(BTW, I have a brand new interpretation of Aristoxenus that
is yet to be added to this page, in which all of his math 'works'
for the first time.)

Aristoxenus insisted on characterizing pitch as occurring
on a continuum, from the lowest notes, which he called 'relaxed',
to the highest, which he called 'tense'. He used poles of
srting tension rather than string length because he did not make
use of any rational mathematics in his music-theory. (Today
we normally think of musical ratios as those between *frequencies*;
in ancient times it was always ratios of *string length*.)

The term which I translate 'relaxed', _malakon_, is usually
given in English (as in Partch's _Genesis_) as 'soft', but
this fails to convey the sense of opposing poles on a continuum,
hence the reason I prefer 'relaxed'.

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:

> But if one insists on *accuracy* for rational tunings, I think
> my system or the variant of it by Daniel Wolf is the clearest
> way to do it.

Yes. For strict JI this is definitely a good way to go. Traditional
notation for the chain of fifths, and a fixed set of alterations for
other notes. This will make it obvious whether two scales are
transpositions of one another or not. In the Johnston notation,
determining this can take a good hour of head-scratching (as I found
out when I tried looking at the score for his String Quartet #4).

I strongly urge any 11- (or even 17-) limit strict JI composer to at
least listen to his or her music in 72-tET. Besides the well-known
names, it was Tenney's choice as a practical realization of strict
JI. Daniel Wolf mentioned that Partch (an 11-limit strict JI
composer) could not hear any deviation from his system in Erv
Wilson's 41-tET tubolong -- and 41-tET has much larger errors in the
11-limit than 72-tET. Though conceivably a piece of 11-limit strict
JI music might wander through a series of 225:224s, which vanish in
72-tET, the majority of progressions in 11-limit strict JI, played at
a reasonable pace, will be aurally identical in 72-tET.

Even if there were no advantage to compatibility with 12-tET, I would
use a 72-part octave to measure 11-odd-limit intervals. That's
because it's the simplest equal division in which none of the 11-
limit diamond ratios (or 11-odd-limit intervals) deviates by 1/4 of a
division or more. In other words, if we associate with any equal
temperament a "gray zone" extending from n+.25 to n+.75 between steps
n and n+1, none of the diamond ratios falls into the gray zone. Thus
it not only corresponds to what 72-tET _is_; it also completely
avoids what 72-tET _is not_.

The corresponding lowest "gray-less" ETs for various odd limits are:

odd limit ET
7 4

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:

> But if one insists on *accuracy* for rational tunings, I think
> my system or the variant of it by Daniel Wolf is the clearest
> way to do it.

I strongly urge any 11- (or even 17-) limit strict JI composer to at
least listen to his or her music in 72-tET. Besides the well-known
names, it was Tenney's choice as a practical realization of strict
JI. Daniel Wolf mentioned that Partch (an 11-limit strict JI
composer) could not hear any deviation from his system in Erv
Wilson's 41-tET tubolong -- and 41-tET has much larger errors in the
11-limit than 72-tET. Though conceivably a piece of 11-limit strict
JI music might wander through a series of 225:224s, which vanish in
72-tET, the vast majority of progressions in 11-limit strict JI,
played at
a reasonable pace, will be aurally identical in 72-tET.

Even if there were no advantage to compatibility with 12-tET, I would
use a 72-part octave to measure 11-odd-limit intervals. That's
because it's the simplest equal division in which none of the 11-
limit diamond ratios (or 11-odd-limit intervals) deviates by 1/4 of a
division or more. In other words, if we associate with any equal
temperament a "gray zone" extending from n+.25 to n+.75 between steps
n and n+1, so that exactly half of the pitch continuum is gray, none
of the diamond ratios falls into the gray zone. Thus
it not only corresponds to what 72-tET _is_; it also completely
avoids what 72-tET _is not_.

The lowest "gray-less" ETs for various odd limits are:

odd limit ET

7 31
9 41
11 72
13 270
15 494
17 3395
19-21 8539

It is fortuitous that 72-tET happens to be a simple subdivision of
our present 12-tone system, and has been proposed as a standard even
by microtonalists who don't give a damn about ratios or consonance,
but just want to keep "emanicipating the dissonances". 50/72 octave
is also an amazingly good approximation of Phi. I can see no other
tuning system that could so brilliantly fulfill the desires of such
radically different musical schools of thought.

Finally, the "big daddy" of CPS scales, the 4)8 {1,3,5,7,9,11,13,15}
hebdomekontany (70 tones), maps consistently to 72-tET, since "gray-
less-ness" is equivalent to Hahn level-2 consistency, and all the
intervals in the CPS are at most level-2 in the 11-odd limit. You can
see this mapping on the third page of
http://www.anaphoria.com/starrswitch.html; the hebdomekontany has
been used by Stephen James Taylor among others . . .

🔗manuel.op.de.coul@eon-benelux.com

🔗Joseph Pehrson <pehrson@pubmedia.com>

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
>
http://www.egroups.com/message/tuning/15288
> Joe,
>
> My book goes into some detail about Johnston's notational
> system, and how mine differs from it - take a look at the
> 'Beyond 19' chapter.
>

Hi Monz!

Thanks, I forgot about that... I will "check it out..." (Well, I
won't have to "check it out"... I OWN it!)

> And for the record, I haven't looked into Graham's notation
> yet, I'm glad Paul agrees with me, and I totally endorse
> 72-tET for its practicality.
>
> But if one insists on *accuracy* for rational tunings, I think
> my system or the variant of it by Daniel Wolf is the clearest
> way to do it.
>

As I mentioned, it does seem that those people PARTICULARLY focused
on JI wouldn't want the 12-tET as the basic grid... Personally, it's
OK by me, but JI is only ONE(ONE) of the things I do...

Joe

🔗Joseph Pehrson <pehrson@pubmedia.com>

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:

http://www.egroups.com/message/tuning/15289

>
> This scale was first recorded by Ptolemy (100s AD, Alexandria).
> Ptolemy's theories were rendered into Latin by Boethius c. 505,
> and the scale was advocated by Zarlino in the mid-1500s, writing
> in Italian, as the basis of musical harmony. Keep in mind that
> Zarlino was also the first theorist to extend consideration of
> harmonic units from dyads to triads; all other harmonic theoretical
> speculation before him dealt with *intervals* (dyads).
>
> As far as I know, the introduction of this tuning into the
> German literature occurred in the early 1600s by Lippius and
> then soon after by Rene Descartes (the philosopher).
>

Thanks so much, Monz, for the background on this! I knew we could
rely on you!!! In fact, I believe now I recall Harry Partch talking
about the Ptolemy connection to this scale in his "History of
Intonation..." It has, indeed, be around awhile!

>
> > P.S. And what does "systonon" mean (??)
>
>
> That's supposed to be 'syntonon'. It's Greek for 'tense'.
>
> Use of the term originated with the theories of Aristoxenus.
> See my paper (not finished and in need of much editing...sorry)
> at:
>
> http://www.ixpres.com/interval/monzo/aristoxenus/318tet.htm
>

This is, without a doubt, one of the most "exhaustive" studies I have
ever seen! Monz, when did you learn Greek?? As preparation for this
study?? The whole thing is incredibly impressive, and, I must
confess, I have not studied the entire yet... just some of it.

May I be forgiven, though, if I found the following paragraph rather
humorous (??):

[Litchfield 1988, p 64-65]
The crux of this proof is whether the fifth is a full perfect
>fifth. In concept, there is no reason to admit this interval as
>anything but a fifth; in practice, however, it does not sound like a
>fifth. Had Aristoxenus performed this proof, the would surely have
>heard the discrepancy, as would any other musicician used to
>manipulationg the monochord. Nevertheless, the discrepancy between
>theory and practice appears not to have bothered Aristoxenus. He
>changed nothing in his treatise because of it. ... Thus, the concept
>alone must have been Aristoxenus's concern, and he appears not as an
>empiricist but as an conceptualist or conceptual idealist.

Perhaps he was one of our first "conceptual artists??"

________ ___ __ _ _
Joseph Pehrson

🔗Joseph Pehrson <pehrson@pubmedia.com>

🔗Bill Alves <ALVES@ORION.AC.HMC.EDU>

I have recently written a couple of pieces in which acoustic performers are
asked to tune to a tape (or disc, rather) computer-generated accompaniment
in extended JI. I have opted for a system in which I use symbols to
indicate *approximate* deviations from 12TET. Though the latest of these
pieces hasn't been performed yet, it is my belief that it will be a
practical way for performing musicians to approach a score like this with a
minimum of special training in notation. Here are the symbols I use:

+/- slightly sharp/flat -- deviation by about a 12th
tone
up arrow/down arrow rather sharp/flat -- deviation by about a sixth tone
half sharp/backwards flat about a quarter-tone sharp/flat
#/b (conventional flat and sharp) about a semitone

These can be combined, usually the +/- or arrows with conventional flats or
sharps. This gives me the same number of pitches as 72TET, though in JI. I
don't need a precise notation (like Johnston's or Wolf's or Monzo's)
because the players are usually tuning to a reference pitch sounding in the
computer accompaniment. They just need to know where approximately they
need to aim for in relation to the pitches they're used to playing. Because
of the consistency of 72TET in relation to JI, the approximation is usually
good to within a few cents.

Bill

^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
^ Bill Alves email: alves@hmc.edu ^
^ Harvey Mudd College URL: http://www2.hmc.edu/~alves/ ^
^ 301 E. Twelfth St. (909)607-4170 (office) ^
^ Claremont CA 91711 USA (909)607-7600 (fax) ^
^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗David Finnamore <daeron@bellsouth.net>

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Monz <MONZ@JUNO.COM>

--- In tuning@egroups.com, "Joseph Pehrson" <pehrson@p...> wrote:

> http://www.egroups.com/message/tuning/15302
>
> --- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
>
> > http://www.egroups.com/message/tuning/15289
> >
> > ...
> >
> > That's supposed to be 'syntonon'. It's Greek for 'tense'.
> >
> > Use of the term originated with the theories of Aristoxenus.
> > See my paper (not finished and in need of much editing...sorry)
> > at:
> >
> > http://www.ixpres.com/interval/monzo/aristoxenus/318tet.htm
> >
>
> This is, without a doubt, one of the most "exhaustive" studies
> I have ever seen! Monz, when did you learn Greek?? As
> preparation for this study?? The whole thing is incredibly
> impressive, and, I must confess, I have not studied the entire
> yet... just some of it.

Thanks very much for the compliments, Joe. About this same
paper, I was honored by what I consider the greatest compliment
anyone's ever given me about my work: John Chalmers (of all
people) said: 'your scholarship is breathtaking'.

In fact, I learned Greek *while* working on this paper... wanting
to translate Aristoxenus *literally* into English was the catalyst.
I can't really say I did actually study or learn Greek... I just
figured out enough of it to translate these passages. (And,
I hasten to add, John helped me over some of the rough spots.)

And please keep in mind that this paper is *very* sloppy as
it stands right now, and is also missing my latest interpretation
of Aristoxenus's tetrachord divisions, which I consider to be a
real breakthru. So it's probably less than 1/2 developed so far,
definitely still in the incubation stages.

>
> May I be forgiven, though, if I found the following paragraph
> rather humorous (??):
>
> [Litchfield 1988, p 64-65]
> The crux of this proof is whether the fifth is a full perfect
> > fifth. In concept, there is no reason to admit this interval as
> > anything but a fifth; in practice, however, it does not sound
> > like a fifth. Had Aristoxenus performed this proof, the would
> > surely have heard the discrepancy, as would any other musicician
> > used to manipulationg the monochord. Nevertheless, the
> > discrepancy between theory and practice appears not to have
> > bothered Aristoxenus. He changed nothing in his treatise
> > because of it. ... Thus, the concept alone must have been
> > Aristoxenus's concern, and he appears not as an empiricist
> > but as an conceptualist or conceptual idealist.
>
> Perhaps he was one of our first "conceptual artists??"

Yes, Joe, I can see why you find it humorous. But seriously,
I would say that's not too far off the mark of the conclusion
Litchfield reached. Litchfield would certainly call Aristoxenus
a 'conceptual theorist', even if 'artist' is pushing it a bit.
(We only have his lectures as written down by students, no
'evidence' of compositions or even any musical illustrations.)

As a matter of fact, this little discrepancy described by
Litchfield still occurred in the interpretation I've published
on my webpage, and is exactly what still bothered me about it.
In my new interpretation, the tuning method described by
Aristoxenus ends with an interval at the end here which is
699 cents, which is audibly indistinguishable from a 'perfect 5th'.
That's the main reason why I think my latest interpretation is
the correct one.

Since many of you may by now be wondering, here's my new
interpretation in a very small nutshell:

Aristoxenus tuned the 'fixed notes' of his tetrachords 'by concords',
that is, as 'perfect 4ths' by ear, which is more-or-less synonymous
with making them 4:3 ratios. Then, he divided the string-lengths
of each of these 4:3s into 5 equal parts, so that a '4th' really
*does* equal '2 and a half tones'.

The resulting 'Greater Perfect System' has lots of 3- and 5-limit
ratios, and - something which stunned me - several of 17- and
19-limit which are *identical* to the ones proposed by Ganassi
in 1543 to produce the 'ancient genera' (_Regula Rubertina_,
chapter 4... Ganassi is also in my book.)

This interpretation of Aristoxenus also produces a plethora of
super-particular ratios between scale degrees, as reflected in
the tunings of all later Greek theorists, such as Eratosthenes,
Didymus, Ptolemy, and Aristides Quintilianus; I consider this
to be yet more 'evidence' in support of my new interpretation.

I'm going to be pretty much offline for the next month, as I
have to travel to Philadelphia for a few weeks. (Really
disappointed to miss Microthon by such a narrow margin!...)
So I won't have time to put the new Aristoxenus stuff up on
the site for a while yet... but if anyone wants to do the
calculations, all you need is the info above.

(Hopefully I'll be able to stop in and visit my NYC colleagues
while on the east coast.)

-monz
http://www.ixpres.com/interval/monzo/homepage.html

🔗Monz <MONZ@JUNO.COM>

--- In tuning@egroups.com, Bill Alves <ALVES@O...> wrote:

> http://www.egroups.com/message/tuning/15306
>
> This gives me the same number of pitches as 72TET, though in JI.
> I don't need a precise notation (like Johnston's or Wolf's or
> Monzo's) because the players are usually tuning to a reference
> pitch sounding in the computer accompaniment. They just need to
> know where approximately they need to aim for in relation to the
> pitches they're used to playing. Because of the consistency of
> 72TET in relation to JI, the approximation is usually good to
> within a few cents.

This is pretty much exactly how Ezra Sims summed up his methods
to me. And what he said also echoed Paul Erlich's statements
in his recent post analyzing 72-tET great abilities to 'mimic'
high-limit JI: Sims has made computer versions of his pieces
in both 37-limit JI and 72-tET, and says that unless hard pressed,
he can't tell the difference. His notation is entirely 72-tET,
but because the harmonies are theoretically based on JI and
summation and difference tones, the players can use their ears
to adjust their intonation and get it quite accurate.

Perhaps this is also the place to say that, while I've had
much criticism for Ben Johnston's notational system, Johnston
developed it gradually out of exactly the same principles:
symbols that represent relationships that the performers can
*hear*, and therefore 'get right'. It's simply his use
of a 2-factor scale (prime-factors 3 and 5) for the basic
symbol set that I see as causing the problems with it. It's
an entirely logical system, but more complicated than it had
to be.

-monz
http://www.ixpres.com/interval/monzo/homepage.html

JI notation

🔗graham@microtonal.co.uk

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Joseph Pehrson <josephpehrson@compuserve.com>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Joseph Pehrson <josephpehrson@compuserve.com>

🔗Paul H. Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Joseph Pehrson <josephpehrson@compuserve.com>

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Monz <MONZ@JUNO.COM>

🔗Monz <MONZ@JUNO.COM>

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗manuel.op.de.coul@eon-benelux.com

🔗Joseph Pehrson <pehrson@pubmedia.com>

🔗Joseph Pehrson <pehrson@pubmedia.com>

🔗Joseph Pehrson <pehrson@pubmedia.com>

🔗Joseph Pehrson <pehrson@pubmedia.com>

🔗Joseph Pehrson <pehrson@pubmedia.com>

🔗Bill Alves <ALVES@ORION.AC.HMC.EDU>

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗David Finnamore <daeron@bellsouth.net>

🔗Paul Erlich <PERLICH@ACADIAN-ASSET.COM>

🔗Monz <MONZ@JUNO.COM>

🔗Monz <MONZ@JUNO.COM>