Defining Just intonation (was: Graylessness and limit)

Paul Erlich:

>>I was under the impression that this list is concerned
>> with tuning as it relates to _music_ -- correct me if I'm wrong.

Monz:

>Of course I wouldn't say that you're 'wrong', Paul. But to me
>this list isn't *exclusively* about music.
>
>The example that comes immediately to mind is the ancient idea
>of the 'Music of the Spheres': the idea that relationships of
>orbital periods, distances, etc., between various heavenly bodies
>are in 'harmonic' proportions.

No, but in so far as it's about tuning, it's about tuning as it relates to
_music_, not the tuning of carburettors and not the "tuning" of planetary
orbits, unless there is actually some relationship (as there is in your
case) between the "tuning" of planetary orbits and musical tuning.

Keep watching, any day now we'll have the 'Music of the Carburettors'. ;-)

Monz:

>I've been reading the discussion of various flavors of minor
>triad with interest, and just thought I'd add that one which
>I really liked in a particular context was the 64:75:96 which
>I used at the climactic point in the melody of _3 Plus 4_:
>
>http://www.ixpres.com/interval/monzo/3plus4/3plus4ji.mid
>
>This chord makes its first appearance at 20 seconds into the tune.
>
>(An mp3 version is available on the Tuning Punks site.)
>
>When I first uploaded this JI-retuned version at the very end
>of last year, I discussed this chord a bit on this list; check
>the archives. The point I found most interesting was that
>75/64 is much closer in frequency to 7/6 than to 19/16 - both
>of which I tried first - but it sounds far more different (to my
>ears) from the former than from the latter. I never did figure
>out how to explain that.

Are you claiming that 64:75:96 is a justly intoned chord? I don't think so.
The reason it sounds more like 16:19:24 than 6:7:9 is because both 16:19:24
and 64:75:96 are on the harmonic entropy plateau between the valley chords
6:7:9 and 1/(6:5:4).

ratio cents diff(cents)
----------------------------------
6:7 266.9 valley
7.7
64:75 274.6
9.0
(5+6phi): 283.6 middle of plateau
(6+7phi) 13.9
16:19 297.5
18.1
5:6 315.6 valley

ratio cents diff(cents)
----------------------------------
4:5 386.3 valley
18.1
19:24 404.4
18.0
(4+7phi): 422.5 middle of plateau
(5+9phi) 4.9
75:96 427.4
7.7
7:9 435.1 valley

Margo Schulter:

>Please let me begin by saying that I consider just intonation (JI) in
>the most comprehensive classic sense to mean exactly what David
>Beardsley has said: tuning by integer ratios.

I hope you can see, as David B has, that this definition is completely
useless because, for example, it would count the Hammond organ [brilliant
example Paul E] as JI when it clearly is not. One would only need to listen
to it to be convinced of that.

From The Shorter Oxford English Dictionary on Historical Principles.
Just
/Mus./ in /just interval/, etc. : Harmonically pure; sounding perfectly in
tune 1811.

The important point is that whether or not an interval or chord is 'justly
intoned', is not a mathematical property, it's about how something
_sounds_! I find it ironic that I, a mathematical non-musician am telling a
bunch of musicians this.

>Of course, I also agree with Dave Keenan that there are different
>_kinds_ of JI intervals and styles, and that a ratio such as 3:2, for
>example, has a different kind of "justness" from a ratio of
>intermediate complexity such as 21:17, or one of impressive complexity
>such as 12544:9801 (81:64 plus two 896:891 commas).

No. 12544:9801 isn't any kind of JI interval. Not because it has large
numbers in it, but because it isn't close enough to any ratio with small
enough numbers. It might by accident be a consequence of some JI tuning,
but there is no way it could be called a just interval. You only have to
listen to it, and then try varying the pitch of one of the notes slightly
in both directions to hear that there is nothing "just" about it.

>Attempts to set some limit on the size of integer ratios used in JI
>seem to me both needlessly restrictive and futile, but certainly we
>can recognize the fact that more complex JI intervals have a quality
>much like those of tempered (irrational) ratios surrounding them; and
>similarly that some tempered intervals are "virtually just."

If an interval "has a quality much like those of tempered (irrational)
ratios surrounding it", this is precisely what defines it as not just!

>For example, when I tune 24-note Pythagorean or "Xeno-Gothic" (the
>latter term implying a neo-Gothic stylistic context), I certainly
>consider this a "JI tuning," a description including all the ratios
>and intervals of the tuning, however simple (2:1, 3:2, 4:3, 9:8) or
>complex.

It may be a "JI tuning" (although most are not) but that does not make all
the intervals it contains "JI intervals".

>At the same time, I recognize that the interval of 16 fifths up, for
>example, 43046721:33554432 (~431.28 cents), is "just" in the sense
>that it is an integer ratio derived from pure fifths, but from another
>point of view a "virtually tempered" approximation of 9:7, being
>around 3.80 cents narrow of this simple ratio.

The _only_ reason to consider 43046721:33554432 as just is because it is
within 3.8 cents of 7:9. But it might better be called quasi just. 3.8
cents is getting a little far away from such a complex ratio as 7:9.

>The idea of "tuning by integer ratios," of course, may carry various
>implications for various people and styles of music, for example:
>
>(1) Using the simplest and purest possible ratios for stable
>concords in a given style;
>
>(2) Preferring epimores or superparticular ratios for intervals
>making up a tetrachord, for example (e.g. 9:8-8:7-28:27);
>
>(3) Seeking to maximize the simplicity or "suavity" of certain
>unstable intervals and combinations as well as stable ones,
>e.g. 4:5:6:7 in an 18th-century setting (Euler), or 7:9:12
>in a neo-14th-century setting.
>
>Possibly one might use capitalization to suggest a difference in
>nuance between a "just" interval (any integer ratio) and a Just
>interval (an interval with a _small_ integer ratio readily tuneable by
>ear by a locking in of partials).

No. This would be a complete departure from history. Motivated apparently
by mathematics (or numerology) rather than art or aesthetics.

just = Just = pure = tuneable by ear

>Also, JI systems feature what we might regard as three types of
>intervals:
>
>(1) "Simple" JI intervals a la Dave Keenan, or the "valleys"
>of Paul Erlich's harmonic entropy, where locking in of
>partials occurs, with Paul's suggested upper limit of
>complexity by the point for a ratio a:b where a*b=105.
>
>(2) "Intermediate" JI intervals such as 14:11 or 13:11 where
>combination tones may give ratios something of a
>"quasi-Just" quality (I leave this as an open question).
>
>(3) "Complex" JI intervals such as the regular Pythagorean
>major and minor thirds at 81:64 and 32:27, as well as
>a variety of large-integer ratios in more recent JI
>systems (e.g. LaMonte Young).

No. There is no comparison between 81:64 and 32:27 as typically used and
LaMonte Young's massive sine-wave clusters. I can accept that the Dream
House is a form of JI. It depends on extremely accurate tuning and uses a
huge number of simultaneous tones to match its high ratios. It matters very
much how accurately those tones are tuned. It matters very little how
accurately 81:64 and 32:27 are tuned in Gothic music. 81:64 and 32:27 are
not just intervals in this context.

>A typical JI system may have some intervals from each of these three
>categories, which is to say that _some_ just intervals are also Just
>or "pure."

I'm sorry I _just_ can't accept that definition. Here again:

Just
/Mus./ in /just interval/, etc. : Harmonically pure; sounding perfectly in
tune 1811.

Regards,
-- Dave Keenan
http://dkeenan.com

David C Keenan wrote,

> The important point is that whether or not an interval or chord is
'justly intoned', is not a mathematical property, it's about how
something _sounds_! I find it ironic that I, a mathematical
non-musician am telling a bunch of musicians this.

Oh come now! I respect the type of distinctions your tying to make
Dave, but all this assumes NO musical context or aesthetic/artistic
intention whatsoever... and this is a big mistake to my mind. I mean
if I play "Happy Birthday" in octaves at the piano is it not justly
intoned? Is it now "just intonation" music! Or how about a piece or
body of music composed in strict JI that zips along at a rapid enough
clip as to render even sizable tuning errors all but a moot point...
is this music any less JI than some long drawn-out, easy as pie
musical demonstration of the same justly intoned intervals?

If you hope to make these types of distinctions truly meaningful in a
big picture real world sense, then I really think you'll need to bring
a bit more than a nice little definition from The Shorter Oxford
English Dictionary on Historical Principles to the table.

--Dan Stearns

Hi Dave

Mathematically stated, the problem is that Just intonation, defined in terms of tunable to pure ratios by ear, is
intransitivie (or fuzzily transitive perhaps).

So Margo is suggesting we use Just for the intransitive term, and just (uncapitalised) for its transitive closure.

That seems an eminently sensible suggestion to me.

(I too am a mathematician)

Robert

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:

> 12544:9801 isn't any kind of JI interval. Not because it has
large
> numbers in it, but because it isn't close enough to any ratio with
small
> enough numbers. It might by accident be a consequence of some JI
tuning,
> but there is no way it could be called a just interval.

Accidents are not necessarily responsible for high-term ratios. They
can be a necessary mathematical consequence of a (arguably) JI
tuning. 12544:9801, or (7^2*2^8):(11^2*3^4), does seem like an
extreme example whether you look at its odd limit or view it as a
member of a rather extensive harmonic lattice. But I think Margo
made
some very strong points. I like her suggested trio of just interval
types, though it may need some terminological refinement. Maybe
they're not types of just intervals but only types of intervals
arising from Just tuning systems. (We do need to be careful not to
be
inclusive to the point of making terms meaningless.) But I find them
to be helpful categories, especially in light of her neo-Gothic
theoretical treatises.

Between her and Dave K., I'm wondering whether we need a new term
(Oh, no! Not again! }:- ). How should we categorize an interval
that
is a consequence of a JI tuning, yet sounds like a tempered interval?
How can it be called a tempered interval if both of its tones are
members of a JI tuning? If it's neither tempered nor just, what is
it? "Rational" comes to mind, but it's a bit broad.

This is similar to the issue of 4+ digit ETs, which I lampooned
recently. Some people might have thought I was making fun of Paul E.
That was not the case. The idea of such large ETs conjured up some
very strange images in my mind and I just couldn't resist sharing
them. But clearly, there is a sound(!) theoretical reason to speak
of an eight thousand some-odd ET, even though there is no practical
musical use for an instrument with tens of thousands of notes, nor
for one with intervals of tiny fractions of a cent between adjacent
tones.

Similarly, there is good reason, in the right theoretical tuning
context, to speak of high-term rational intervals, even when they are
audibly indistinguishable from certain irrational intervals and/or
lower term rational intervals. They are demanded by some consistent
rational systems, while the intervals they sound like are explicitly
excluded. As Joe Monzo cleverly reminds us, all roads lead to n^0.
But you still have to choose a road.

When David Beardsley recently spoke of boldly using higher members of
the harmonic series, I couldn't see at first how it could require
courage to do so. But in a sense he's right. The higher you go, the
greater danger that your audience will get lost in the harmonic
terrain. It's good to know where the pitfalls lie but I'm not sure
it's possible to predict with certainty the exact point of no return.
A great composer may be able to keep listeners well oriented by any
number of means. If s/he means to confuse them, then I suppose the
intervals must be considered tempered for all practical purposes.

David J. Finnamore

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:

> Margo Schulter:
>
> >Please let me begin by saying that I consider just intonation (JI)
in
> >the most comprehensive classic sense to mean exactly what David
> >Beardsley has said: tuning by integer ratios.
>
> I hope you can see, as David B has, that this definition is
completely
> useless because, for example, it would count the Hammond organ
[brilliant
> example Paul E] as JI when it clearly is not.

While the Hammond Organ example [ref. msg. 15815] does demonstrate
that not all rational tunings are Just tunings, it does not
constitute
proof that Just Intonation only applies to simple ratios. The reason
is that the Hammond ratios were chosen to approximate 12 EDO as
closely as possible with the technology at hand, which happened to be
tone wheels. As a set of Just ratios, it is an arbitrary set. Even
though its highest term contains only two digits, it does not provide
a Just tuning. It is, in the truest sense, an accidental rational
tuning.

OTOH, consider the 24 member chain of pure fifths, 1:1 3:2 9:8 ...
94143178827:68719476736, of which our friend Margo Schulter is fond.
As a set of Just ratios, it is a systematic set. Every member is
related to at least one other member, and in all but two cases to two
other members, by 3:2. Even though its highest term contains 11
digits, it does provide a Just tuning.

Consider this set of ratios:

8:5 5:3 12:7 16:9 13:10 7:4 11:6

Is that a Just tuning? Odd limit = 13. But, no, it's arbitrary.
It's meaningless as a set of Just ratios.

Perhaps by the word "tuning," in the quote at the top of this post,
Margo intended to imply (as seems likely) organizing pitches into a
system for the purpose of aiding the theorist, composer, and listener
in discovering meaning in (or imposing meaning on) the pitch
continuum. If so, then the Hammond Organ tuning would not have to be
included in her definition of Just Intonation, and I would agree with
her on it.

David Finnamore

David Finnamore wrote:

> When David Beardsley recently spoke of boldly using higher members of
> the harmonic series, I couldn't see at first how it could require
> courage to do so. But in a sense he's right. The higher you go, the
> greater danger that your audience will get lost in the harmonic
> terrain. It's good to know where the pitfalls lie but I'm not sure
> it's possible to predict with certainty the exact point of no return.
> A great composer may be able to keep listeners well oriented by any
> number of means. If s/he means to confuse them, then I suppose the
> intervals must be considered tempered for all practical purposes.

Confused? Consider this: JI (and microtonality in general)
broadens the resources available to composers. I don't
see anybody releasing albums called "Music for Conservative
Mathematicians".

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

David Finnamore wrote:

> Consider this set of ratios:
>
> 8:5 5:3 12:7 16:9 13:10 7:4 11:6
>
> Is that a Just tuning? Odd limit = 13.

Yep.

> But, no, it's arbitrary.
> It's meaningless as a set of Just ratios.

Only because you can't find a pattern. But there IS a pattern in
this collection of numbers. Ratios.

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

--- In tuning@egroups.com, David Beardsley <xouoxno@v...> wrote:
>
> Confused? Consider this: JI (and microtonality in general)
> broadens the resources available to composers. I don't
> see anybody releasing albums called "Music for Conservative
> Mathematicians".
>
> --
> * D a v i d B e a r d s l e y
> * 49/32 R a d i o "all microtonal, all the time"
> * http://www.virtulink.com/immp/lookhere.htm

David has here touched on a very important topic for me. Which can
best be put as a question: Is the microtonal music we make "tuning
theory" (or the realization thereof), or is it "music". Ultimately,
and most importantly, I believe it's the latter that must always take
precedence. Microtonal tuning theory does not good music make. And
one must face a huge "alienation factor" with the consumer of this
kind of niche market, when the focus is on the theory.

When I did the first draft of the liner notes for my
CD "Metamorphose", I didn't even mention the tuning system(s)
employed. I noticed that Kraig Grady also mentions little about his
tunings in his CDs. I wouldn't attempt to speak for Kraig's purposes,
but for myself, I just know that to get folks to appreciate this
music, it can be crucial to consider the packaging as well as the
content.

What do you put forth with microtonal music that draws in the
uninitiated? It's an important question to ask in the face of many
cultural facts, such as: symphonies disbanding for lack of support,
high school music programs being discontinued, classical music
concert and CD sales at an all time low, and the ever deeper
entrenchment of the big conglomo "content controlling" major record
labels, who have such a power grip on the airwaves; distributing
their own brand of "audio valium" - this is happening all over the
country. What do microtonal composers have to contribute in these
changing times? Part of the answer for me is beautiful music. When I
play a Kraig Grady CD, I'm not concerned with fondling the packaging,
I'm swept away by the breathtaking solos (Santur? Mt.Meru! - goose
pimples every time!), and ensemble compositions - which really
require little explanation, because the music's communicating it's
intention without the programmatic notes. Even looking from this side
of the fence, I'm listening for the music to be good - to have some
kind of compelling quality that will make me want to hear it again
and again - and I become more interested in the theory later. I
believe it's important to consider that most folks listen to music on
a purely emotional level. Considering this, what foot should one put
forward? The math or the music? Perhaps there's a balance one must
strike between the two. I think we all would like to see the level of
cultural assimilation of microtonality that would allow the theory to
disappear into the background of our musical culture, in the manner
that we've assimilated our current 12 tET, but we face a difficult
reality.

Can I get an Amen-a-?

Jacky Ligon

ligonj@northstate.net wrote:

> I
> believe it's important to consider that most folks listen to music on
> a purely emotional level.

That's right. Consider popular music (includes bubblegum, rap, metal,
dance and so on...). It's not popular because the general public
knows anything about tuning issues.

> Can I get an Amen-a-?

Yep.

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

--- In tuning@egroups.com, David Beardsley <xouoxno@v...> wrote:
> David Finnamore wrote:
>
> > Consider this set of ratios:
> >
> > 8:5 5:3 12:7 16:9 13:10 7:4 11:6
> >
> > Is that a Just tuning? Odd limit = 13.
>
> Yep.
>
> > But, no, it's arbitrary.
> > It's meaningless as a set of Just ratios.
>
> Only because you can't find a pattern. But there IS a pattern in
> this collection of numbers. Ratios.

Why wouldn't that argument apply to the Hammond ratios?

--- In tuning@egroups.com, ligonj@n... wrote:
> I
> believe it's important to consider that most folks listen to music
on
> a purely emotional level. Considering this, what foot should one
put
> forward? The math or the music? Perhaps there's a balance one must
> strike between the two. I think we all would like to see the level
of
> cultural assimilation of microtonality that would allow the theory
to
> disappear into the background of our musical culture, in the manner
> that we've assimilated our current 12 tET, but we face a difficult
> reality.
>
>
> Can I get an Amen-a-?
>
> Jacky Ligon

Believe me, the vast majority of people who listen to and enjoy my
music, even my small microtonal repertoire, have no idea that there's
any math behind it whatsoever. And that's the way it should be. The
emotional, or rather aesthetic level, is paramount, and the math is
just a tool to aid in one aspect of the ascent. Sometimes this list
gets into math for its own sake (q.v. Monz talking about tunings
apart from music), which is fine if some of us need a "math fix" once
in a while, but Jacky, I'm with you on this . . . you don't see
equations of statics inscribed in 8-foot lettering on the Golden Gate
Bridge, do you?

Paul Erlich wrote:

> > Only because you can't find a pattern. But there IS a pattern in
> > this collection of numbers. Ratios.
>
> Why wouldn't that argument apply to the Hammond ratios?

The Hammond organ was obviously designed to be a 12tet instrument.
Finding one or more than a few that are not in tune does'nt make
it a JI instrument. It sounds like the Hammond had/has bad resolution.
Does this make my Proteus 2 a JI instrument? No.

Sometimes stretching logic to it's limit just makes one look silly.

db
--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

On Sat, 25 Nov 2000 06:22:42 -0000, "David Finnamore"
<daeron@bellsouth.net> wrote:

>Between her and Dave K., I'm wondering whether we need a new term
>(Oh, no! Not again! }:- ). How should we categorize an interval
>that
>is a consequence of a JI tuning, yet sounds like a tempered interval?
> How can it be called a tempered interval if both of its tones are
>members of a JI tuning? If it's neither tempered nor just, what is
>it? "Rational" comes to mind, but it's a bit broad.

If it's a good approximation of a consonant just interval, or an interval
in a tempered scale, it would be reasonable to consider it a tempered
interval, like the 8192/6561 of Pythagorean tuning, which is within 2 cents
of 5/4, or the 63/50 of 7-limit JI, which is barely more than 1 cent away
from the 12-TET equivalent.

Even though 63/50 isn't in itself a simple ratio that can be tuned
accurately by ear or heard as consonant (except as a distant approximation
of 5/4), it can easily turn up in a 7-limit scale. I ran across it in a
12-note 7-limit scale that I've been playing with.

1/1 21/20 9/8 6/5 63/50 4/3 7/5 3/2 63/40 42/25 9/5 189/100 2/1

But if I ran across it in the following scale:

1/1 89/84 55/49 44/37 63/50 4/3 99/70 3/2 73/46 37/22 98/55 17/9 2/1

I'd conclude that it was meant to be a tempered interval. This seemingly
haphazard set of pitches is in fact a good approximation of 12-TET. I don't
think there'd be much point in calling this scale "just", although
technically it might be so if the definition of "just" is broad enough.

So the question is what to call 63/50 in that 7-limit JI scale I was
talking about. In this case, all the notes of the scale are related by
factors of 3/2, 6/5, or 7/5 to other notes in the scale. Besides the 63/50,
this scale contains a 112/75, which is less than 8 cents (224/225) away
from 3/2. It certainly seems reasonable to call the 112/75 a tempered 3/2,
even though it resulted from the combination of simple 7-limit intervals.
The 63/50 is a little more problematic, since it's not really a very good
approximation to 5/4, but since it's only a 12-note tuning, I think it
would be perceived as a temered 5/4. In a larger 7-limit tuning, the
126/125 difference between 5/4 and 63/50 might be significant enough to
consider them as distinct intervals.

--
see my music page ---> ---<http://www.io.com/~hmiller/music/music.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

--- In tuning@egroups.com, David Beardsley <xouoxno@v...> wrote:
> Paul Erlich wrote:
>
> > > Only because you can't find a pattern. But there IS a pattern in
> > > this collection of numbers. Ratios.
> >
> > Why wouldn't that argument apply to the Hammond ratios?
>
> The Hammond organ was obviously designed to be a 12tet instrument.
> Finding one or more than a few that are not in tune does'nt make
> it a JI instrument. It sounds like the Hammond had/has bad
resolution.
> Does this make my Proteus 2 a JI instrument? No.
>
> Sometimes stretching logic to it's limit just makes one look silly.

There was nothing silly about David Finnamore's arguments. He was
pointing out to you and Dave Keenan that the way you construct a just
tuning has much more to do with the justness's musical relevance than
simply the size of the numbers involved. And I agree that a bunch of
random 13-limit pitches with no logical relation with one another,
just like Finnamore's example, are as much (or as little) JI as the
Hammond tuning. It may be JI by definition, but it fails to serve a
composer who is looking to expliot the audible advantages/unique
qualities of JI.

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:

> http://www.egroups.com/message/tuning/15836
>
> Monz:
>
> > I've been reading the discussion of various flavors of minor
> > triad with interest, and just thought I'd add that one which
> > I really liked in a particular context was the 64:75:96 which
> > I used at the climactic point in the melody of _3 Plus 4_:
> >
> > http://www.ixpres.com/interval/monzo/3plus4/3plus4ji.mid
> >
> > This chord makes its first appearance at 20 seconds into the tune.
> >
> > (An mp3 version is available on the Tuning Punks site.)
> >
> > When I first uploaded this JI-retuned version at the very end
> > of last year, I discussed this chord a bit on this list; check
> > the archives. The point I found most interesting was that
> > 75/64 is much closer in frequency to 7/6 than to 19/16 - both
> > of which I tried first - but it sounds far more different (to my
> > ears) from the former than from the latter. I never did figure
> > out how to explain that.
>
> Are you claiming that 64:75:96 is a justly intoned chord? I don't
> think so.
> The reason it sounds more like 16:19:24 than 6:7:9 is because both
> 16:19:24 and 64:75:96 are on the harmonic entropy plateau between
> the valley chords 6:7:9 and 1/(6:5:4).
>
> ratio cents diff(cents)
> ----------------------------------
> 6:7 266.9 valley
> 7.7
> 64:75 274.6
> 9.0
> (5+6phi): 283.6 middle of plateau
> (6+7phi) 13.9
> 16:19 297.5
> 18.1
> 5:6 315.6 valley
>
>
> ratio cents diff(cents)
> ----------------------------------
> 4:5 386.3 valley
> 18.1
> 19:24 404.4
> 18.0
> (4+7phi): 422.5 middle of plateau
> (5+9phi) 4.9
> 75:96 427.4
> 7.7
> 7:9 435.1 valley

Thanks, Dave, for the enlightening explanation as to why the
75:64 sounded more like the 19:16 (which is farther away) than
the 7:6 (which is closer).

But as far as the question of whether there's any validity to
my claim that 64:75:96 is 'a justly-intoned chord', I'll have
to disagree with you and attempt to defend that claim. This
retuned version of _3 Plus 4_ was most definitely meant to be
a 'JI' version, and all of the other chords I use in this tune
have much simpler proportions. I simply wanted a particular
sound at this particular point in the tune which (to my ears)
demanded this particular tuning. The overall context of the
whole tune, in this version, is definitely JI.

Certainly, the numbers in the proportions of this one chord
happen to be much larger than the numbers in the proportions
of all the other chords, but that (IMO) does not invalidate
the 'JI-ness' of this one chord.

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

Paul Erlich wrote:

> There was nothing silly about David Finnamore's arguments. He was
> pointing out to you and Dave Keenan that the way you construct a just
> tuning has much more to do with the justness's musical relevance than
> simply the size of the numbers involved. And I agree that a bunch of
> random 13-limit pitches with no logical relation with one another,
> just like Finnamore's example, are as much (or as little) JI as the
> Hammond tuning. It may be JI by definition, but it fails to serve a
> composer who is looking to expliot the audible advantages/unique
> qualities of JI.

The scale in question is JI no matter how badly it's constructed.
Quit trying to reinvent the wheel.

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

> [Me:]
> > The higher you go [in the harmonic series], the
> > greater danger that your audience will get lost in the harmonic
> > terrain. It's good to know where the pitfalls lie but I'm not sure
> > it's possible to predict with certainty the exact point of no return.
> > A great composer may be able to keep listeners well oriented by any
> > number of means. If s/he means to confuse them, then I suppose the
> > intervals must be considered tempered for all practical purposes.
>
> Confused? Consider this: JI (and microtonality in general)
> broadens the resources available to composers.

Agreed. In fact, I've been quite an evangelist for that idea in my
little music community for about three years.

> I don't
> see anybody releasing albums called "Music for Conservative
> Mathematicians".

That's very funny! Although it may actually be a valid niche
market. ;-)

But here "confused" = "lost in the harmonic terrain," not something
like "confused about what JI is for." Using higher (odd, especially
prime) harmonic relationships in a tuning requires increasing
compositional effort to keep the listener from being confused about
the harmonic identity of the pitches. This is true whether or not
the listener knows anything about the mathematical theory behind
it. That assumes that the human psycho-acoustical apparatus has a
built in harmonic template. And if it doesn't, what's the point of
sticking rigidly to JI at all?

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

David Beardsley wrote:

> > 8:5 5:3 12:7 16:9 13:10 7:4 11:6
> >
> > Is that a Just tuning? Odd limit = 13.
>
> Yep.
>
> > But, no, it's arbitrary.
> > It's meaningless as a set of Just ratios.
>
> Only because you can't find a pattern. But there IS a pattern in
> this collection of numbers. Ratios.

The fact that they're all ratios is a common characteristic, not a
pattern. To put it another way, is it possible to construct a
logical tuning system that would yield this as a complete set? It
can't be proven that it isn't because that's a universal negative.
But it seems highly unlikely to me. I chose them arbitrarily.
Well, not quite - I chose to list 7 items, and specifically excluded
choices that would tend to result in anything resembling a
recognizable scale. That's harder to do than it sounds, a testimony
to the inherent musical vigor of low-ratio JI. Once you concoct a
set like this, though, what you've got is a "tuning" that is almost
inevitably going to yield "music" that is little more than a jumble
of meaningless tones, provided that the style of music is one in
which musical interest relies on tonal relationships.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

--- In tuning@egroups.com, Herman Miller <hmiller@I...> wrote:
> On Sat, 25 Nov 2000 06:22:42 -0000, "David Finnamore"
> <daeron@b...> wrote:
[snip]
> > How should we categorize an interval
> >that
> >is a consequence of a JI tuning, yet sounds like a tempered

interval? [snip]
> If it's a good approximation of a consonant just interval, or an

interval
> in a tempered scale, it would be reasonable to consider it a

tempered
> interval, like the 8192/6561 of Pythagorean tuning, which is within

2 cents
> of 5/4

How about, "if it is used as another just interval..."? When I
first

wrote the sentence above, I put "yet sounds and/or functions like a

tempered interval?" But I pulled that part of it because function
is

the point here, I think. If it acts like a 5/4, it is a 5/4. But
in

the context of a tuning and compositional style that excludes
5-limit

functions, 8192/6561 may function instead as the pure 3:2 below

4096/2187. This is where the scale member 8192/6561 is different
from

the interval 8192:6561. In that case, I think it is a just interval

in its own right. But, wishing to respect the viewpoints of others

(and recognizing the fallibility of my own), I ask whether we might

refine our terminology to reflect this possibility. IOW, have

separate terms for each of Margo's three categories of "just"

intervals.

> Even though 63/50 isn't in itself a simple ratio that can be tuned
> accurately by ear or heard as consonant (except as a distant

approximation
> of 5/4), it can easily turn up in a 7-limit scale. I ran across it

in a
> 12-note 7-limit scale that I've been playing with.
>
> 1/1 21/20 9/8 6/5 63/50 4/3 7/5 3/2 63/40 42/25 9/5 189/100 2/1
>
> But if I ran across it in the following scale:
>
> 1/1 89/84 55/49 44/37 63/50 4/3 99/70 3/2 73/46 37/22 98/55 17/9 2/1
>
> I'd conclude that it was meant to be a tempered interval. This

seemingly
> haphazard set of pitches is in fact a good approximation of 12-TET.

Great examples! This is exactly what I was saying (or meant to
say!)

about the Hammond tuning (see

http://www.egroups.com/message/tuning/15861). Haphazard rational

tunings are not JI, except, of course, by accident. To state this

from the other side of the coin, JI depends as much on systematic

organization of the pitch continuum as on the use of ratios, IMHO.
Is

this an appropriate refinement of the definition of JI, or am I

stretching it too far?

> So the question is what to call 63/50 in that 7-limit JI scale I was
> talking about. In this case, all the notes of the scale are related

by
> factors of 3/2, 6/5, or 7/5 to other notes in the scale.

For that reason, I would call 63/50 a just interval in this context,

with the caveat that it could easily be used as 5/4 in a composition

using this tuning if the composer were not careful. Or a composer
may

wish to blur the line to artistic effect (the "confusion" in my
reply

to David Beardsely http://www.egroups.com/message/tuning/15897).

> The 63/50 is a little more problematic, since it's not really a very

good
> approximation to 5/4, but since it's only a 12-note tuning, I think

it
> would be perceived as a temered 5/4. In a larger 7-limit tuning, the
> 126/125 difference between 5/4 and 63/50 might be significant enough

to
> consider them as distinct intervals.

Did somebody say "harmonic entropy"? But a composer might keep the

harmonic identity clear by being careful to play the 63/50 only in
its

3:2, 6:5, and/or 7:5 relationships to other members of the scale,
and

not to do such things as invoke the triad 1/1 63/50 3/2, which will

almost certainly be heard as a tempered 4:5:6 chord for all intents

and purposes. This is one way a composer may bravely push on up the

harmonic series without disorienting his listeners in the harmonic

terrain.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

--- In tuning@egroups.com, "Paul Erlich" <PERLICH@A...> wrote:
> --- In tuning@egroups.com, David Beardsley <xouoxno@v...> wrote:
> > Sometimes stretching logic to it's limit just makes one look
silly.
>
> There was nothing silly about David Finnamore's arguments. He was
> pointing out to you and Dave Keenan that the way you construct a
just
> tuning has much more to do with the justness's musical relevance
than
> simply the size of the numbers involved. And I agree that a bunch
of
> random 13-limit pitches with no logical relation with one another,
> just like Finnamore's example, are as much (or as little) JI as the
> Hammond tuning. It may be JI by definition, but it fails to serve a
> composer who is looking to expliot the audible advantages/unique
> qualities of JI.

Bingo. Thank you, Paul, that's what I needed - someone to put it
simply and concisely.

I wasn't sure whether David B. meant me or you (or both) about
looking silly. But if it was me, no hurt/hard feelings here. :-)
It is kind of difficult to carry on a meaningful discussion when the
participants don't all agree on the ground rules, but we'll get
through it. A little chaos in a system can keep things from getting
in a rut.

David Finnamore

[David Finnamore wrote...]

>Consider this set of ratios:
>
>8:5 5:3 12:7 16:9 13:10 7:4 11:6
>
>Is that a Just tuning? Odd limit = 13. But, no, it's arbitrary.
>It's meaningless as a set of Just ratios.

It all depends, David. There's a convention on this list that helps
avoid much confusion -- /'s for pitches, and :'s for intervals.

Writing the above with :'s, as you did, there is indeed an odd limit
of 13. But written with /'s, the odd limit would be much higher,
since we must consider all the intervals between the pitches.

But regarding your point, 'is the tuning just'... I would say that
no tuning is just. Pythagorean tuning, when used in the medieval
style, clearly portraits 3-limit just intonation. Used another way,
it may not. The tonality diamond is capable of all crazy sorts of
dissonance, but Partch and Prent Rodgers have created beautiful
justly-intoned music with it.

Why is just often written with an upper-case "j" on this list? I
find that telling. The term's meaning is quite simple: correct
intonation. Not: correct tuning. All verbs may eventually become
nouns, but let's try to preserve meaning along the way.

-Carl

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

> But as far as the question of whether there's any validity to
> my claim that 64:75:96 is 'a justly-intoned chord', I'll have
> to disagree with you and attempt to defend that claim. This
> retuned version of _3 Plus 4_ was most definitely meant to be
> a 'JI' version, and all of the other chords I use in this tune
> have much simpler proportions. I simply wanted a particular
> sound at this particular point in the tune which (to my ears)
> demanded this particular tuning. The overall context of the
> whole tune, in this version, is definitely JI.

But _that chord_ isn't audibly different from a non-JI chord,
whatever the context . . .
>
> Certainly, the numbers in the proportions of this one chord
> happen to be much larger than the numbers in the proportions
> of all the other chords, but that (IMO) does not invalidate
> the 'JI-ness' of this one chord.

How do you feel about the Hammond organ, then?

--- In tuning@egroups.com, David Beardsley <xouoxno@v...> wrote:

> The scale in question is JI no matter how badly it's constructed.
> Quit trying to reinvent the wheel.

What wheel am I trying to reinvent??? I'm just trying to point out
some important distinctions David Finnamore brought up that can help
validate useful assessments such as your own that the Hammond organ
is not JI. I personally have no interest in what the definition is,
only to clarify and make logically sound the usage that is relevant
to musicians such as yourself, Finnamore, etc. . . but apparantly
this touches some kind of nerve . . . I think I'm going to excuse
myself from this "Defining Just Intonation" discussion henceforth.

On Sun, 26 Nov 2000 12:16:57 -0600, "David J. Finnamore"
<daeron@bellsouth.net> wrote:

>the point here, I think. If it acts like a 5/4, it is a 5/4. But
>in
>
>the context of a tuning and compositional style that excludes
>5-limit
>
>functions, 8192/6561 may function instead as the pure 3:2 below
>
>4096/2187. This is where the scale member 8192/6561 is different
>from

I meant 8192:6561. I've always used fractions for intervals and I keep
forgetting that there's a different convention on this list.

>Did somebody say "harmonic entropy"? But a composer might keep the
>
>harmonic identity clear by being careful to play the 63/50 only in
>its
>
>3:2, 6:5, and/or 7:5 relationships to other members of the scale,
>and
>
>not to do such things as invoke the triad 1/1 63/50 3/2, which will
>
>almost certainly be heard as a tempered 4:5:6 chord for all intents
>
>and purposes. This is one way a composer may bravely push on up the
>
>harmonic series without disorienting his listeners in the harmonic
>
>terrain.

I've been trying to avoid the 63:50 in contexts that make it sound like a
tempered 5:4, but I don't hesitate to use the 112:75 as a 3:2. Of course,
one interval that can be heard as a tempered interval doesn't make this a
tempered scale any more than quarter-comma meantone is a JI scale for
having just 5:4's.

Of course, since I'm doing this on the DX7II, I'm really using a subset of
1024-TET that approximates a just scale :) ... which ultimately has to be a
rational approximation of 1024-TET since it's all done digitally, so it's a
JI scale after all! :)

--
see my music page ---> ---<http://www.io.com/~hmiller/music/music.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

Margo Schulter wrote

>>At the same time, I recognize that the interval of 16 fifths up, for
>>example, 43046721:33554432 (~431.28 cents), is "just" in the sense
>>that it is an integer ratio derived from pure fifths, but from another
>>point of view a "virtually tempered" approximation of 9:7, being
>>around 3.80 cents narrow of this simple ratio.

Dave Keenan wrote

>The _only_ reason to consider 43046721:33554432 as just is because it is
>within 3.8 cents of 7:9. But it might better be called quasi just. 3.8
>cents is getting a little far away from such a complex ratio as 7:9.

43046721:33554432 could be rendered in just intonation as a 3/2 to the
fifth 15 fifths up [14348907/8388608] but could not stand as a just
interval to 1/1.

Even most digital synthesisers cannot produce platonic just intonation. I
believe Marion Mckosky [author of the Fastrak sequencer] developed a
circuit divider that achieved exact ratio divisions rather than greater or
lesser degrees of resolution to just intervals [really just very fine equal
temperament, although on Kyma and some synths you might have to wait a
while to hear the beats]

Justin White

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Hi all,

I'm sorry if my previous post in this thread seemed somewhat blunt and
unsympathetic. I wrote it in a hurry as I was going away for a few days and
it was in response to posts that were already many days old. My apologies.

----------------------------------
Non-just intervals in Just tunings
----------------------------------

Several people have expounded on one of the distinctions that Margo made.
High integer ratios that arise by chaining low ones versus the low ones
themselves. I agreed with the distinction. In fact I thought it was
obvious. But I could not agree with the terminology Margo proposed.

One reason I could never accept Margo's suggestion to call these big "J"
Just and little "j" just is because I (and others) already use big "J" to
to distinguish it from little "j" meaning "merely" [at least that's why
_I_ do it Carl, although I don't disagree with your reason].

Justness is indeed intransitive [Thanks Robert Walker]. Take a Just
diminished triad B:D:F where B:D and D:F are both tuned 5:6. Merely because
B:D is just and D:F is just, doesn't mean that B:F is just. 25:36 is not
something anyone could tune by ear with ordinary timbres. However it makes
perfect sense to refer to the whole chord as just, because there is no
remaining freedom to justly intone the B:F independently of the B:D and
D:F. The same goes for whether or not to call a scale or tuning just.

The fact that the B:F tuning is a medium integer ratio is entirely
incidental. Dave Finamore, I should have written "incidental" when I wrote
"accidental".

We usually have two types of interval in a Just tuning: Just intervals and
non-just intervals. A non-just interval isn't necessarily a tempered
interval, since "tempered" implies a deliberate attempt to compromise. The
25:36 above is the result of a complete refusal to compromise on the two
5:6's, but it still isn't a Just interval (although it might be in the
context of a 25:27:30:33:36 chord). A tempered diminished chord might take
account of the proximity of 25:36 to 5:7 or 7:10.

So I don't see a need for a new term, unless you count "non-just" as new.
They are simply non-just intervals that occur in just tunings. I realise
though that I have often, in the past, overloaded the terms consonance and
dissonance for this purpose. As Dave Beardsley pointed out, an 8:13 (or
even an 8:15?) might be justly intoned, but you'd hardly call them
consonances.

Dave Finnamore, I fail to understand how the Hammond organ tuning is not an
"organizing [of] pitches into a system for the purpose of aiding the
theorist, composer, and listener in discovering meaning in (or imposing
meaning on) the pitch continuum." It fails to be a JI tuning by failing to
be JI, not by failing to be a tuning. Whereas your collection of 13 limit
just intervals fails to be a tuning because there is nothing to relate them
to each other. Unless you mean them to be stacked on top of each other in
the order given, in which case it is a tuning, but it stands or falls as JI
on whether 10:13 can be tuned by ear or not (which I severely doubt). In
any case it isn't useful or interesting, but that's not what I'm trying to
define.

The Hammond pitch ratios all relate to a common 1/1 at 320 Hz. It clearly
is a tuning (by any definition of tuning that I am aware of). The fact that
the 1/1 is not included in the tuning does not disqualify it from being JI;
witness Wilson's CPS's. What disqualifies it is the fact that the just
intervals that it _does_ contain (octaves), have not locked it up
completely. Other intervals could be justly intoned without making any of
the existing just intervals non-just.

A Pythagorean tuning with Just octaves and fifths is JI because it is fully
committed or locked up. One must back-track and break one or more of the
just fifths before one can go forward again to say a 5-limit JI tuning.

Monz, you haven't told me why you think 64:75:96 is justly intoned. I'll
tell you why I think it isn't. It's not because the numbers are too big.
It's because I set up my synth with unadorned sawtooth waves, split the
keyboard and transposed the low part up an octave, played the middle note
on the low half so I could move it, and only it, around with the pitch bend
joystick, and found absolutely nothing that could distinguish the 64:75:96
from other pitches in the general vicinity. I could easily find the justly
intoned 6:7:9 and 1/(6:5:4). I had to have it pretty loud and listen pretty
carefully before I could find anything that _might_ be the 16:19:24 (or was
it an approximate 11:13|11:14?). But absolutely nothing special happens at
64:75:96. So how can you say it is justly intoned? Maybe you were afraid I
would say your piece was not JI because it contained one non-just chord. I
hope it is clear that is not the case.

-----------------------------------------------
Just not small integer - small integer not just
-----------------------------------------------

I think that any tuning that has as many as possible of its intervals
(dyads) resulting in coincidence of partials, is a just tuning (or JI
tuning), even when it only has that property for some particular inharmonic
timbre (a la Sethares). In that case we need to say that the tuning is
"Just for timbre x". This is a case where a Just tuning has absolutely
nothing to do with integer ratios, small, large or otherwise.

When we say a tuning is Just without qualification, we mean it is Just for
typical harmonic timbres. But a tuning that is Just for harmonic timbres
will also be Just for inharmonic timbres and sine waves, not because of
coinciding partials, but because of the periodicity effect or the "harmonic
series template" in the brain.

The Hammond organ tuning is an example of a non-just tuning that uses
precise medium integer ratios (no integer greater than 85). A Setharian
inharmonic spectrum scale is an example of a Just tuning that doesn't use
integer ratios at all. So "Just" and "limited-integer-ratio" are categories
that significantly overlap, but neither is wholly contained in the other.

Dave Beardsley, the Hammond organ tuning is not a poorly-tuned 12-tET at
all. One would not find many 12-tET acoustic instruments (or JI acoustic
instruments for that matter) tuned so accurately as to have errors less
than 1 cent in all intervals.

------------------------------------
"Rational tuning" considered useless
------------------------------------

If "rational tuning" means "a tuning based on ratios", I hope we can see
that this is a purely mathematical property and has no relationship to any
audible quality of a tuning, or the intervals or chords available in it.
One cannot tell by listening, or even by measuring with the most
sophisticated instruments possible, whether a tuning is rational or not.
That distinction belongs only to the platonic realm of pure mathemtics. In
the "real" world no frequency ratio can ever be said to be rational or
irrational. It would take an infinite time of counting cycles before one
could decide.

As such, I find the concept of rational tuning to be of no interest
whatsoever. Limited-integer-rational is of interest, but does not
correspond to Just.

Incidentally, for me a "scale" is primarily a melodic entity, a tuning may
support many scales or a tuning may be a tuning of a single interval or
chord. But "scale" and "tuning" are often used interchangeably and that's
ok. Does anyone have other distinctions between scale and tuning?

---------------------------------
How does context affect justness?
---------------------------------

Dan Stearns, I can accept that some aspects of musical context, e.g. timbre
and harmonic context, can make a difference as to whether some medium
integer ratio interval is considered justly intoned. The test is still
whether listeners can tell when you vary its tuning by a few cents in that
context. I don't accept that an otherwise non-just interval can become Just
merely by playing it for a very short time. I think it is understood that
there is some lower limit on the duration of the listening test, of the
order of a second, or the maximum decay time of the instrument.

But I certainly can't accept that anyone's _intention_ (performer, composer
or scale creator) can make a difference to whether or not an interval is
justly intoned. Give me an example where you think it should.

Rest assured that I do not think Happy Birthday played in octaves on a
12-tET instrument qualifies as JI music. Of course every vertical interval
played is justly intoned, but the underlying scale is not, because it has
freedom to justly intone many other intervals that it does not. But I'm not
currently trying to improve our definition of "JI music", but only "JI
tuning" (and hence Just scale, Just chord, Just interval). The most basic
one is Just interval (there's no such thing as a just pitch, except in
relation to other pitches). The other definitions will be built on top of
this.

-----------------------
Nice little definitions
-----------------------

Does anyone have any other "nice little" definitions of "Just interval"
from other respected authorities? Can anyone suggest why the Oxford
Dictionary definition is dated 1811?

I still maintain that the Justness of an interval is primarily determined
by listening while a pitch is varied slightly one way and the other. There
is a gray area, but many intervals are clearly just or non-just by this
criterion.

Then we go to the mathematics to try to find simple formulae that let us
predict the justness of an interval. We do this mathematical modelling so
we don't have to correlate the opinions of 50 expert listeners every time
we want to know if some tuning will be just or not.

Unfortunately, and understandably, this modeling has led to people
(generally non-mathematicians) being seduced by the numbers to such a
extent that they can flatly make statements like "Just intonation is simply
tuning by integer ratios", a nice little definition which on closer
examination is seen to be utterly useless.

Regards,
-- Dave Keenan
http://dkeenan.com

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:
> [David Finnamore wrote...]
>
> >Consider this set of ratios:
> >
> >8:5 5:3 12:7 16:9 13:10 7:4 11:6
> >
> >Is that a Just tuning? Odd limit = 13. But, no, it's arbitrary.
> >It's meaningless as a set of Just ratios.
>
> It all depends, David. There's a convention on this list that helps
> avoid much confusion -- /'s for pitches, and :'s for intervals.

Oh - I may have had that backwards. And I see that I didn't use the
convention the way I understand it. As I meant to show the ratios of
a tuning, each to 1/1, I should have written it as 8/5 5/3 etc.,
right? The way I wrote it is the usual way to show the intervals
between the tones of the scale? Thankfully context made it clear.

> But regarding your point, 'is the tuning just'... I would say that
> no tuning is just.

That's an interesting point of view. Then what do you call tunings
composed of rational intervals? Especially those intended to provide
maximal just intonation of individual intervals?

> Why is just often written with an upper-case "j" on this list?

Someone (Margo?) recently proposed guidelines for when to capitalize
it. I go by gut instinct, but my supposed guideline is, if I'm using
it in a name (e.g., "this piece is in Just Intonation") I capitalize
it. Otherwise (e.g., "the players used just intonation") I don't.
Above I did not follow that guideline. :-/

> The term's meaning is quite simple: correct
> intonation. Not: correct tuning.

"Correct intonation" is a matter of context. In many contexts,
rational intonation is incorrect intonation. I think any definition
is going to have to make some reference to ratios to be accepted by
most of us.

> All verbs may eventually become
> nouns, but let's try to preserve meaning along the way.

Agreed about preserving meaning (once we determine the meaning). But
I used it as an adjective both times. How would it be used as a
verb?
"See Dick. See Dick just. Just, Dick, just!" I can't think of a
noun form, either. "Set the just on the table and keep your hands
where I can see them!" With an added -ly it can be an adverb,
though:
"Dick justly intoned the final 7/4."

Send your sentences for diagramming to:

David Finnamore

Dave K.:

Nice post! You make a lot of sense when you're not it a hurry. :-)

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:

> Dave Finnamore, I fail to understand how the Hammond organ tuning
is not an
> "organizing [of] pitches into a system for the purpose of aiding the
> theorist, composer, and listener in discovering meaning in (or
imposing
> meaning on) the pitch continuum."

12 EDO is, of course, but, *as a set of ratios*, the Hammond table is
not because the ratios were not chosen to do so. They are merely an
approximation of 12 EDO. It's a fine distinction but an important
one to understanding where I was going with my proposed enhancement to
the definition of JI proposed by Margo.

> It fails to be a JI tuning by
failing to
> be JI, not by failing to be a tuning.

?! True. I didn't say it failed to be a tuning. It fails to be the
tuning it intends to approximate, though not by enough to matter.
But I didn't say that either.

> Whereas your collection of 13 limit
> just intervals fails to be a tuning because there is nothing to
relate them
> to each other.

Exactly. That was the essence of my point. Now here we reach the
crux of the matter:

> but it stands or falls as JI
> on whether 10:13 can be tuned by ear or not (which I severely
doubt).
[snip]
> Monz, you haven't told me why you think 64:75:96 is justly intoned.
I'll
> tell you why I think it isn't. It's not because the numbers are too
big.
> It's because I [snip] found absolutely nothing that could
distinguish the 64:75:96
> from other pitches in the general vicinity.
[snip]
> If "rational tuning" means "a tuning based on ratios", I hope we
can see
> that this is a purely mathematical property and has no relationship
to any
> audible quality of a tuning, or the intervals or chords available
in it.
> One cannot tell by listening, or even by measuring with the most
> sophisticated instruments possible, whether a tuning is rational or
not.
> That distinction belongs only to the platonic realm of pure
mathemtics. In
> the "real" world no frequency ratio can ever be said to be rational
or
> irrational. It would take an infinite time of counting cycles
before one
> could decide.
[Good point!] [snip]
> The test is still
> whether listeners can tell when you vary its tuning by a few cents
in that
> context.
[snip]
> I still maintain that the Justness of an interval is primarily
determined
> by listening while a pitch is varied slightly one way and the
other. There
> is a gray area, but many intervals are clearly just or non-just by
this
> criterion.

Of course! Why didn't I see this before? This is the old Aristotle
vs. Plato argument. This is a matter which hasn't been solved in
2600 years, and we're not likely to solve in on the Tuning List! LOL.

But my main problem with making the question "Can it be tuned by
ear?"
the main criterion is that it's not possible to define at precicely
what point intervals become untunable by ear. You have a definition
of "just" that varies from person to person, from time to time, from
place to place, from instrument to instrument - in short an
unreliable
definition, and therefore one which can have no universal application.

My step-grandfather probably couldn't have tuned 2:1 properly on a
pair of strings, never mind 5:4; he couldn't carry a tune in a
proverbial bucket. In college, as a music major, I would almost
certainly have tuned a major third much closer to 400 cents than to
5:4, even as a singer in a barbershop quartet. I can now tune 11:4
quite reliably by ear using stable, harmonic timbres, something I
could not have done two years ago. Yet last week, while recording a
piece with multiple diatonic pennywhistle parts, I had a great deal
of trouble telling whether certain passages were sharp or flat to
each
other. Very disconcerting. One thing I noticed, though not for the
first time, is that tuning can sound radically different on
headphones than through speakers.

I know you will probably never agree with me that terms like "JI"
need a platonic definition, and I will not likely concede that they
need a practical one. But can you really place all the weight on
something as mutable as human hearing?

David Finnamore

"David J. Finnamore" wrote:

> But here "confused" = "lost in the harmonic terrain," not something
> like "confused about what JI is for." Using higher (odd, especially
> prime) harmonic relationships in a tuning requires increasing
> compositional effort to keep the listener from being confused about
> the harmonic identity of the pitches. This is true whether or not
> the listener knows anything about the mathematical theory behind
> it. That assumes that the human psycho-acoustical apparatus has a
> built in harmonic template. And if it doesn't, what's the point of
> sticking rigidly to JI at all?

A non conservative approach to composition that hopefully results in
fresh new music.

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

First I wrote:

> > The scale in question is JI no matter how badly it's constructed.

And immediately after that I wrote:

> > Quit trying to reinvent the wheel.

Paul Erlich wrote:

> What wheel am I trying to reinvent??? I'm just trying to point out
> some important distinctions David Finnamore brought up that can help
> validate useful assessments such as your own that the Hammond organ
> is not JI. I personally have no interest in what the definition is,
> only to clarify and make logically sound the usage that is relevant
> to musicians such as yourself, Finnamore, etc. . . but apparantly
> this touches some kind of nerve . . . I think I'm going to excuse
> myself from this "Defining Just Intonation" discussion henceforth.

Unfortunately Paul only read the 2nd part.

--
* D a v i d B e a r d s l e y
* 49/32 R a d i o "all microtonal, all the time"
* http://www.virtulink.com/immp/lookhere.htm

David C Keenan wrote,

> I'm sorry if my previous post in this thread seemed somewhat blunt
and unsympathetic <SNIP>

Hi Dave, as usual, many good points/ideas... I'll try not to belabor
my own point of view and feelings here as I think I've already laid
them out, and I personally don't really care much one way or the
other, as I think JI is a pretty hearty (if somewhat ill-defined)
entity already, and I just don't think that's about to change much one
way or the other.

To me "just intonation" is a wide-ranging but generally easily
understood umbrella term in which all the technical minutia and
musical (and with just intonation often ideological) practice rest. We
can better define, hone and understand the minutia, but the entity
just intonation is already alive in well in the popular lexicon so to
speak, and any sort of a "recall" at this point seems kind of
foolhardy to my mind.

> How does context affect justness?

I remember Wendy Carlos commenting on "Partch's folly". The idea being
that someone so interested and motivated by aural causation goes on to
compose a body of work that 'zips right by', and somehow fails to best
exploit this phenomena. To my mind this is the logical road that your
line of reasoning leads to, and to be rather blunt myself, I'd have to
say that I really do find this rather offensive -- folly indeed!

Partch of course knew darn well what he was doing and did what damn
well interested him. Composers and artist always need and I think
deserve a bit of imaginative elbowroom, a bit of "elision"... and
especially so when a great body of achievement backs this up.

I think there will always be friction between those who desire clear,
well-defined agendas and precise inventories of all the addressable
moving parts, and those who work at music/art in the usual way -- a
personally arrived at conglomeration of what seems right... of what
works.

--Dan Stearns

--- Paul Erlich:

> http://www.egroups.com/message/tuning/15907
>
> --- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
>
> > But as far as the question of whether there's any validity to
> > my claim that 64:75:96 is 'a justly-intoned chord', I'll have
> > to disagree with you and attempt to defend that claim. This
> > retuned version of _3 Plus 4_ was most definitely meant to be
> > a 'JI' version, and all of the other chords I use in this tune
> > have much simpler proportions. I simply wanted a particular
> > sound at this particular point in the tune which (to my ears)
> > demanded this particular tuning. The overall context of the
> > whole tune, in this version, is definitely JI.
>
> But _that chord_ isn't audibly different from a non-JI chord,
> whatever the context . . .

While I have to agree with that statement, Paul, I still
think that context plays a very important part in whether
or not one *audibly recognizes* something to be in JI tuning.

When I listen to this piece, in this tuning, this particular
chord defninitely sounds different from a 12-tET chord to me:
the narrowness of the 64:75 'minor 3rd' (= ~275 cents)
is quite distinctive. Of course, that's not to say that
it doesn't resemble many other non-JI chords. Maybe it's
the case that my mind refuses to accept it as anything other
than JI, simply because I know that 64:75 is a 5-limit ratio
and I know how it fits into the 5-limit lattice, and my
foreknowledge affects my perception.

> >
> > Certainly, the numbers in the proportions of this one chord
> > happen to be much larger than the numbers in the proportions
> > of all the other chords, but that (IMO) does not invalidate
> > the 'JI-ness' of this one chord.
>
> How do you feel about the Hammond organ, then?

I would say that the Hammond organ tuning is so close to
12-tET that it doesn't approximate any JI tuning well enough
to imply JI. It's a totally different case from _3 Plus 4_,
where everything else in the song is recognizable, to me
at least, as 11-limit JI.

=============

Dave Keenan:

> http://www.egroups.com/message/tuning/15913
>
> As Dave Beardsley pointed out, an 8:13 (or even an 8:15?)
> might be justly intoned, but you'd hardly call them
> consonances.

I would. I almost always treat both of these intervals
as consonances in my JI pieces, and find that as chord
members they help to produce beautiful chords that I
think of as 'consonant'.

> Monz, you haven't told me why you think 64:75:96 is
> justly intoned. I'll tell you why I think it isn't.
> It's not because the numbers are too big. It's because
> I set up my synth with unadorned sawtooth waves, split
> the keyboard and transposed the low part up an octave,
> played the middle note on the low half so I could move
> it, and only it, around with the pitch bend joystick,
> and found absolutely nothing that could distinguish
> the 64:75:96 from other pitches in the general vicinity.
> I could easily find the justly intoned 6:7:9 and
> 1/(6:5:4). I had to have it pretty loud and listen
> pretty carefully before I could find anything that
> _might_ be the 16:19:24 (or was it an approximate
> 11:13|11:14?). But absolutely nothing special happens
> at 64:75:96. So how can you say it is justly intoned?
> Maybe you were afraid I would say your piece was not
> JI because it contained one non-just chord. I hope
> it is clear that is not the case.
>

Dave, I'm glad you took all the trouble to create a
listening experiment in order to make a more perceptive
observation about this discussion. Beyond the three
tunings I tried for this chord when retuning the piece
from 12-tET, I haven't gone into this kind of depth on
this question. Perhaps an elaboration on my methods
and reasons for choosing 64:75:96 as the tuning for
this triad would be illuminating.

Of course the two outer pitches of the triad are simply
a 2:3 ratio. That part was easy. The trial-and-error
concerned only the tuning of the middle note.

I must admit that I chose 64:75 because I was thinking
in terms of 5-limit when searching for the right tuning
of the 'minor 3rd'. Certainly I could have found other,
higher-prime- or -odd-limit ratios which are very close
in pitch to 64:75 and which would have worked, but I
simply didn't try them.

The retuning of this chord began with the knowledge
that I wanted a 'darker' sound for this chord than
the one provided by the 12-tET version of the minor
triad.

I tried 6:7 for the 'minor 3rd' first, and while
*harmonically* it gave me the 'dark' quality I was
looking for for this particular chord, I didn't like
the way it sounded in the context of the piece, mainly
because it seemed to be out-of-tune with the pitches
in the preceding chord, and 'pain' in the melodic
movement disturbed me. This triad gives a 'minor 3rd'
on the bottom of 6:7 = ~267 cents and a 'major 3rd'
on top of 7:9 = ~435 cents.

So next I tried 16:19. This seemed melodically
smoother but gave a 'minor 3rd' which was too wide
to give me the 'dark' harmonic quality I was seeking;
in short, it resembled the 12-tET version too much.
This triad gives a 'minor 3rd' of 16:19 = ~298 cents
and a 'major 3rd' of 19:24 = ~404 cents.

So, knowing that 64:75 was pretty close in size to the
discarded 6:7, I gave it a shot, and bingo!, it gave
exactly the sound I was looking for: the 'dark' harmonic
quality without the melodic 'pain'. This triad gives a
'minor 3rd' of 64:75 = ~275 cents and a 'major 3rd'
of 25:32 = ~427 cents.

Now, to analyze some other options:

The triad you notate as 11:13|11:14 is 13-prime-limit
and can also be rendered as an otonal proportion by
the numbers 121:143:182. These ratios are indeed rather
close in pitch to the ones I chose, and perhaps would
have worked just as well for me, and at the same time
given lower-integer ratios between the chord-members.
This triad contains a 'minor 3rd' on the bottom of
11:13 = ~289 cents, and a 'major 3rd' on top of
11:14 = ~418 cents.

The 17-prime-limit triad 34:40:51 would have been a good
medium-integer possibility which I would expect would sound
even more like my choice, giving a 'minor 3rd' on the
bottom of 17:20 = ~281 cents and a 'major 3rd' on top
of 40:51 = ~421 cents.

The 23-prime-limit triad 46:54:69 is yet another
possibility, closer still in pitch to my choice, and
giving a 'minor 3rd' of 23:27 = ~278 cents and a
'major 3rd' of 18:23 = ~424 cents.

Of course, with the numbers getting this high, I
could have also given 54:64:81 a go - a regular old
3-limit (= Pythagorean) 'minor triad' with a 'minor 3rd'
of 27:32 = ~294 cents and a 'major 3rd' of 64:81
= ~408 cents. Most likely this would have sounded
pretty similar to my discarded 16:19:24 version.

>
> -----------------------------------------------
> Just not small integer - small integer not just
> -----------------------------------------------
>
>
> I think that any tuning that has as many as possible
> of its intervals (dyads) resulting in coincidence
> of partials, is a just tuning (or JI tuning), even
> when it only has that property for some particular
> inharmonic timbre (a la Sethares). In that case we
> need to say that the tuning is "Just for timbre x".
> This is a case where a Just tuning has absolutely
> nothing to do with integer ratios, small, large or
> otherwise.
>
> When we say a tuning is Just without qualification,
> we mean it is Just for typical harmonic timbres.
> But a tuning that is Just for harmonic timbres will
> also be Just for inharmonic timbres and sine waves,
> not because of coinciding partials, but because of
> the periodicity effect or the "harmonic series
> template" in the brain.
>
> The Hammond organ tuning is an example of a non-just
> tuning that uses precise medium integer ratios (no
> integer greater than 85). A Setharian inharmonic
> spectrum scale is an example of a Just tuning that
> doesn't use integer ratios at all. So "Just" and
> "limited-integer-ratio" are categories that
> significantly overlap, but neither is wholly
> contained in the other.
>

Dave, I think this is a terrific way of making the
kinds of distinctions I like to keep in mind.

On the one hand, it seems to me that the only way
one could create a definitive statement as to
whether a piece is JI or not is to determine
whether or not it *sounds* like it makes use
of 'purely tuned' intervals. On the other hand,
how could this be possible in the case of a composition
which makes much use of 'discordant' intervals?

I think these are important questions worthy of
further debate, because the whole concept of
'just-intonation' is bound up with the desire to
produce - at least some of the time - intervals
which are as 'concordant' as possible.

If a composer (such as myself) is interested in
using what he's always thought of as JI to create
pieces which have lots of chords embodying subtle
shades of dissonance or discordance, is there
any justification at all [pardon the unintended pun]
in *calling* that tuning system JI?

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

--- Paul Erlich:

> http://www.egroups.com/message/tuning/15907
>
> --- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
>
> > But as far as the question of whether there's any validity to
> > my claim that 64:75:96 is 'a justly-intoned chord', I'll have
> > to disagree with you and attempt to defend that claim. This
> > retuned version of _3 Plus 4_ was most definitely meant to be
> > a 'JI' version, and all of the other chords I use in this tune
> > have much simpler proportions. I simply wanted a particular
> > sound at this particular point in the tune which (to my ears)
> > demanded this particular tuning. The overall context of the
> > whole tune, in this version, is definitely JI.
>
> But _that chord_ isn't audibly different from a non-JI chord,
> whatever the context . . .

While I have to agree with that statement, Paul, I still
think that context plays a very important part in whether
or not one *audibly recognizes* something to be in JI tuning.

When I listen to this piece, in this tuning, this particular
chord defninitely sounds different from a 12-tET chord to me:
the narrowness of the 64:75 'minor 3rd' (= ~275 cents)
is quite distinctive. Of course, that's not to say that
it doesn't resemble many other non-JI chords. Maybe it's
the case that my mind refuses to accept it as anything other
than JI, simply because I know that 64:75 is a 5-limit ratio
and I know how it fits into the 5-limit lattice, and my
foreknowledge affects my perception.

> >
> > Certainly, the numbers in the proportions of this one chord
> > happen to be much larger than the numbers in the proportions
> > of all the other chords, but that (IMO) does not invalidate
> > the 'JI-ness' of this one chord.
>
> How do you feel about the Hammond organ, then?

I would say that the Hammond organ tuning is so close to
12-tET that it doesn't approximate any JI tuning well enough
to imply JI. It's a totally different case from _3 Plus 4_,
where everything else in the song is recognizable, to me
at least, as 11-limit JI.

---------------

Dave Keenan:

> http://www.egroups.com/message/tuning/15913
>
> As Dave Beardsley pointed out, an 8:13 (or even an 8:15?)
> might be justly intoned, but you'd hardly call them
> consonances.

I would. I almost always treat both of these intervals
as consonances in my JI pieces, and find that as chord
members they help to produce beautiful chords that I
think of as 'consonant'.

> Monz, you haven't told me why you think 64:75:96 is
> justly intoned. I'll tell you why I think it isn't.
> It's not because the numbers are too big. It's because
> I set up my synth with unadorned sawtooth waves, split
> the keyboard and transposed the low part up an octave,
> played the middle note on the low half so I could move
> it, and only it, around with the pitch bend joystick,
> and found absolutely nothing that could distinguish
> the 64:75:96 from other pitches in the general vicinity.
> I could easily find the justly intoned 6:7:9 and
> 1/(6:5:4). I had to have it pretty loud and listen
> pretty carefully before I could find anything that
> _might_ be the 16:19:24 (or was it an approximate
> 11:13|11:14?). But absolutely nothing special happens
> at 64:75:96. So how can you say it is justly intoned?
> Maybe you were afraid I would say your piece was not
> JI because it contained one non-just chord. I hope
> it is clear that is not the case.
>

Dave, I'm glad you took all the trouble to create a
listening experiment in order to make a more perceptive
observation about this discussion. Beyond the three
tunings I tried for this chord when retuning the piece
from 12-tET, I haven't gone into this kind of depth on
this question. Perhaps an elaboration on my methods
and reasons for choosing 64:75:96 as the tuning for
this triad would be illuminating.

Of course the two outer pitches of the triad are simply
a 2:3 ratio. That part was easy. The trial-and-error
concerned only the tuning of the middle note.

I must admit that I chose 64:75 because I was thinking
in terms of 5-limit when searching for the right tuning
of the 'minor 3rd'. Certainly I could have found other,
higher-prime- or -odd-limit ratios which are very close
in pitch to 64:75 and which would have worked, but I
simply didn't try them.

The retuning of this chord began with the knowledge
that I wanted a 'darker' sound for this chord than
the one provided by the 12-tET version of the minor
triad.

I tried 6:7 for the 'minor 3rd' first, and while
*harmonically* it gave me the 'dark' quality I was
looking for for this particular chord, I didn't like
the way it sounded in the context of the piece, mainly
because it seemed to be out-of-tune with the pitches
in the preceding chord, and 'pain' in the melodic
movement disturbed me. This triad gives a 'minor 3rd'
on the bottom of 6:7 = ~267 cents and a 'major 3rd'
on top of 7:9 = ~435 cents.

So next I tried 16:19. This seemed melodically
smoother but gave a 'minor 3rd' which was too wide
to give me the 'dark' harmonic quality I was seeking;
in short, it resembled the 12-tET version too much.
This triad gives a 'minor 3rd' of 16:19 = ~298 cents
and a 'major 3rd' of 19:24 = ~404 cents.

So, knowing that 64:75 was pretty close in size to the
discarded 6:7, I gave it a shot, and bingo!, it gave
exactly the sound I was looking for: the 'dark' harmonic
quality without the melodic 'pain'. This triad gives a
'minor 3rd' of 64:75 = ~275 cents and a 'major 3rd'
of 25:32 = ~427 cents.

Now, to analyze some other options:

The triad you notate as 11:13|11:14 is 13-prime-limit
and can also be rendered as an otonal proportion by
the numbers 121:143:182. These ratios are indeed rather
close in pitch to the ones I chose, and perhaps would
have worked just as well for me, and at the same time
given lower-integer ratios between the chord-members.
This triad contains a 'minor 3rd' on the bottom of
11:13 = ~289 cents, and a 'major 3rd' on top of
11:14 = ~418 cents.

The 17-prime-limit triad 34:40:51 would have been a
good medium-integer possibility which I would expect
sound even more like my choice, giving a 'minor 3rd'
on the bottom of 17:20 = ~281 cents and a 'major 3rd'
on top of 40:51 = ~421 cents.

The 23-prime-limit triad 46:54:69 is yet another
possibility, closer still in pitch to my choice, and
giving a 'minor 3rd' of 23:27 = ~278 cents and a
'major 3rd' of 18:23 = ~424 cents.

Of course, with the numbers getting this high, I
could have also given 54:64:81 a go - a regular old
3-limit (= Pythagorean) 'minor triad' with a 'minor 3rd'
of 27:32 = ~294 cents and a 'major 3rd' of 64:81
= ~408 cents. Most likely this would have sounded
pretty similar to my discarded 16:19:24 version.

>
> -----------------------------------------------
> Just not small integer - small integer not just
> -----------------------------------------------
>
>
> I think that any tuning that has as many as possible
> of its intervals (dyads) resulting in coincidence
> of partials, is a just tuning (or JI tuning), even
> when it only has that property for some particular
> inharmonic timbre (a la Sethares). In that case we
> need to say that the tuning is "Just for timbre x".
> This is a case where a Just tuning has absolutely
> nothing to do with integer ratios, small, large or
> otherwise.
>
> When we say a tuning is Just without qualification,
> we mean it is Just for typical harmonic timbres.
> But a tuning that is Just for harmonic timbres will
> also be Just for inharmonic timbres and sine waves,
> not because of coinciding partials, but because of
> the periodicity effect or the "harmonic series
> template" in the brain.
>
> The Hammond organ tuning is an example of a non-just
> tuning that uses precise medium integer ratios (no
> integer greater than 85). A Setharian inharmonic
> spectrum scale is an example of a Just tuning that
> doesn't use integer ratios at all. So "Just" and
> "limited-integer-ratio" are categories that
> significantly overlap, but neither is wholly
> contained in the other.
>

Dave, I think this is a terrific way of making the
kinds of distinctions I like to keep in mind.

On the one hand, it seems to me that the only way
one could create a definitive statement as to
whether a piece is JI or not is to determine
whether or not it *sounds* like it makes use
of 'purely tuned' intervals. On the other hand,
how could this be possible in the case of a composition
which makes much use of 'discordant' intervals?

I think these are important questions worthy of
further debate, because the whole concept of
'just-intonation' is bound up with the desire to
produce - at least some of the time - intervals
which are as 'concordant' as possible.

If a composer (such as myself) is interested in
using what he's always thought of as JI to create
pieces which have lots of chords embodying subtle
shades of dissonance or discordance, is there
any justification at all [pardon the unintended pun]
in *calling* that tuning system JI?

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

Sorry about the multiple copies of my response to Paul and Dave K.
eGroups has been slow in posting my reply, and thinking that it
didn't go thru, I sent it several times.

-monz

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:

http://www.egroups.com/message/tuning/15836

tuning by integer ratios.
>
> I hope you can see, as David B has, that this definition is
completely useless because, for example, it would count the Hammond
organ [brilliant example Paul E] as JI when it clearly is not. One
would only need to listen to it to be convinced of that.

I can't restrain myself from posting that I found this whole
discussion of two-digit integer Hammond Organ tuning to be absolutely
hilarious...

_________ ____ __ __
Joseph Pehrson

>Oh - I may have had that backwards. And I see that I didn't use the
>convention the way I understand it. As I meant to show the ratios of
>a tuning, each to 1/1, I should have written it as 8/5 5/3 etc.,
>right?

Yep.

>>But regarding your point, 'is the tuning just'... I would say that
>>no tuning is just.
>
>That's an interesting point of view. Then what do you call tunings
>composed of rational intervals? Especially those intended to provide
>maximal just intonation of individual intervals?

I guess the key is wether we consider a "tuning" to be a list of
numbers only, or a list of numbers with intent. If the latter, "just
tunings" (as opposed to "tempered tunings") might be a good answer... if
the former, "rational tunings" (vs. "irrational tunings") is probably
best.

>>Why is just often written with an upper-case "j" on this list?
>
>Someone (Margo?) recently proposed guidelines for when to capitalize
>it. I go by gut instinct, but my supposed guideline is, if I'm using
>it in a name (e.g., "this piece is in Just Intonation") I capitalize
>it. Otherwise (e.g., "the players used just intonation") I don't.
>Above I did not follow that guideline. :-/

Sounds silly to me.

>>The term's meaning is quite simple: correct intonation. Not: correct
>>tuning.
>
>"Correct intonation" is a matter of context.

Exactly.

>In many contexts, rational intonation is incorrect intonation. I think
>any definition is going to have to make some reference to ratios to be
>accepted by most of us.

Any sound with a clearly defined pitch will oblige itself to rational
intonation. Sethares' work sits at a strange point on this line, but
still obeys it (he has accepted timbres with _slightly_ weaker pitch
definition in exchange for _slightly_ altered rational tunings).

>>All verbs may eventually become nouns, but let's try to preserve meaning
>>along the way.
>
>Agreed about preserving meaning (once we determine the meaning). But I
>used it as an adjective both times. How would it be used as a verb?

By speaking broadly, of course. "Run" was a verb, and later (we assume)
became a noun (a run of cards, etc.). Too often, it seems, the phrase
"Just Intonation" brings to mind a thing, when really, it's an action.
One which has little or nothing to do with wether we use rational or
irrational numbers to describe the frequency relations involved. As the
Hammond example points out (and as Herman Miller reminded, regarding
the resolution of his synth), it doesn't take much to find examples of
numbers from either catagory which impersonate numbers from the other,
as far as the human ear is concerned. And whichever numbers we use, we
face tolerance effects almost immediately, conflicts between place and
periodicity effects left and right, etc. Intonation is tricky stuff, a
bit much for a list of pitches to be expected to capture.

-Carl

Monz wrote:

> So, knowing that 64:75 was pretty close in size to the
> discarded 6:7, I gave it a shot, and bingo!, it gave
> exactly the sound I was looking for: the 'dark' harmonic
> quality without the melodic 'pain'. This triad gives a
> 'minor 3rd' of 64:75 = ~275 cents and a 'major 3rd'
> of 25:32 = ~427 cents.

You tempered it. One could hardly ask for a clearer example of the
concept of tempering. The fact that it's rational is incidental.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

In the discussion between Monz, Dave K., and others, I haven't seen one
very large aspect of tuning addressed: if an interval targets a
particular low integer ratio, yet is allowed to deviate somewhat from it
for purposes of horizontal (melodic) stability, is it still "JI"? How
much deviation is allowed? Zero point zero zero cents?

My own tuning work springs everything, including vertical intervals.
Nothing is "exactly" JI, ever, except in pieces so small or simple as
to have no horizontal tension whatever. (I can, however, stiffen
vertical springs so as to come as arbitrarily close to JI as I wish).

By a strict definition, since every one of my chords is tempered
somewhat from exact JI, none of my chords ARE JI. And yet the chords
are clearly recognizable as far more consonant than 12-tET, precisely
because they target JI.

I have no particular emotional stock in claiming to be "JI", vs.
"micro-tempered JI" or "quasi-JI". But, if we're trying to draw the
boundaries of the meaning of the term Just Intonation, this aspect
should also be addressed.

JdL

--- In tuning@egroups.com, "David J. Finnamore" <daeron@b...> wrote:

> http://www.egroups.com/message/tuning/15963
>
> Monz wrote:
>
> > So, knowing that 64:75 was pretty close in size to the
> > discarded 6:7, I gave it a shot, and bingo!, it gave
> > exactly the sound I was looking for: the 'dark' harmonic
> > quality without the melodic 'pain'. This triad gives a
> > 'minor 3rd' of 64:75 = ~275 cents and a 'major 3rd'
> > of 25:32 = ~427 cents.
>
> You tempered it. One could hardly ask for a clearer example
> of the concept of tempering. The fact that it's rational is
> incidental.

Well, David... I'd tend to agree with you, mostly... but I'm
also willing to believe that I (and other listeners?) can hear
this ratio in this context by means of 'extended reference'
(a la Boomsleiter and Creel), in which case the fact that
it's rational is important.

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

http://www.egroups.com/message/tuning/15913

This was really a fascinating post by Dave Keenan on just
intonation... I've saved it among my important "archives..."
_________ ___ __ _
Joseph Pehrson

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

http://www.egroups.com/message/tuning/15934

> So, knowing that 64:75 was pretty close in size to the
> discarded 6:7, I gave it a shot, and bingo!, it gave
> exactly the sound I was looking for: the 'dark' harmonic
> quality without the melodic 'pain'. This triad gives a
> 'minor 3rd' of 64:75 = ~275 cents and a 'major 3rd'
> of 25:32 = ~427 cents.
>

Joe Monzo's 3+4 is, indeed, an incredible little tune. It can "stay
with" a person for days! The "flatsy" nature of the minor third adds
to the effect... so, I suppose, that associates it in people's minds
with just intonation (??). Surely, it has the same effect as playing
a 7/4 "flatted" in its "pure" state... (??)

But the actual math would make the minor third quite small for JI,
correct... only 275 cents rather than 315, yes (??)

So maybe the association with just intonation has more to do with our
general "flatting" associations with JI tunings (??)

Anyway, it sure works with that piece!!!!
____________ ___ __ _
Joseph Pehrson

----------------------------------
Believe your ears, not the numbers
----------------------------------

Jacky Ligon wrote:

>I believe it's important to consider that most folks listen to music on
>a purely emotional level. Considering this, what foot should one put
>forward? The math or the music? Perhaps there's a balance one must
>strike between the two. I think we all would like to see the level of
>cultural assimilation of microtonality that would allow the theory to
>disappear into the background of our musical culture, in the manner
>that we've assimilated our current 12 tET, but we face a difficult
>reality.
>
>Can I get an Amen-a-?

Amen-a-, Jacky.

Yea verily, that is why I struggle to bring the flock back to the justness
of our forefathers (and mothers), the historical justness that is based on
practical audible qualities (even if they be subliminal to many), not the
completely inaudible, yea even immeasurable, mathematical ones. My children
thou hast strayed from the true path of justness. ;-)

-----------------------
Current mis-definitions
-----------------------

Maybe the following shows where we went wrong. Sorry John. You probably
just got it from someone else.

From John Chalmers, Divisions of the Tetrachord (as quoted in Joe Monzo's
dictionary):
-------------------------------------------------------------------------
Just intonation

Any tuning system which exclusively employs intervals defined by ratios of
integers may be called Just Intonation, though some authors restrict it to
systems whose intervals are derived from the first six overtones, 1, 2, 3,
4, 5, and 6. Such systems are often termed "Five Limit" or "Senary" systems
after Zarlino's "senario" (Partch, 1949, 1974, 1979). The most common
example of such a system is the tuning of the Major Mode as 1/1 9/8 5/4 4/3
3/2 5/3 15/8 and 2/1.

Just Intonation is contrasted to Equal Temperament and Unequal Temperaments
such as Meantone which combine rational with irrational intervals.
-------------------------------------------------------------------------

Systems whose intervals are derived from the first six overtones will
certainly be JI. Our forebears didn't see the need to go higher, but we do.
So lets try saying that _all_ integer ratios result in just intervals.
Seems like an obvious move.

Oops, now we find that we have to call something JI which can't be
distinguished from 12-tET by any _measurement_, let alone by listening to
it. And people start saying things like, "It's the artists intent that
determines whether it is JI or not". Hmm. What if the artist intended to
calculate the right MIDI pitch bends but actually got them all wrong. But
hey everyone still thinks it sounds cool.

This is getting needlessly narcissistic. It sounds like we've backed
ourselves into a corner here.

Ok. Let's try limiting the size of the numbers again. But we can't go too
low, there's the Dream House to consider. And then some bastard shows that
you can still get something that sounds like 12-tET with numbers less than
those used in the Dream House. Hammond bloody organs!

Whada we do now?

Easy. Just go back to the original definition of JI.

------------------------------------------
Definitions that don't even mention ratios
------------------------------------------

-------------------------------------------------------------------------
Just intonation (abbreviated JI)

Intonation means accuracy of pitch or the process of obtaining accurate pitch.

Just intonation is accuracy of one pitch in relation to another pitch, such
that the resulting interval sounds harmonically pure; perfectly in tune;
beatless. Such an interval is called a justly intoned interval or just
interval.

The property of just intonation is also said to be posessed by a musical
work, a body of musical works, or parts of a musical work, such as a
passage, tuning, scale or chord, when these contain a significant number of
just intervals.
-------------------------------------------------------------------------

So what counts as a "significant number" of just intervals for these
various musical entities?

Herman Miller:

>Of course,
>one interval that can be heard as a tempered interval doesn't make this a
>tempered scale any more than quarter-comma meantone is a JI scale for
>having just 5:4's.

Absolutely! It is certainly not necessary that every pitch form a justly
intoned interval with every other pitch. That would limit us to low number
harmonic (or subharmonic) series segments. I agreed with all your other
comments too.

Definition:
-------------------------------------------------------------------------
Justly intoned scale or Just scale (abbreviated JI scale)

A justly intoned scale is is one where every pitch can be reached from
every other pitch by some chain of justly intoned intervals.

Consider a graph with a vertex for each pitch of the scale and an edge for
each justly intoned interval, then a JI scale corresponds to a connected
graph. A non-JI scale has two or more unconnected subgraphs.

If an interval is only perceptible as justly intoned when in some harmonic
context larger than the bare interval, such an interval will count as
justly intoned for the purpose of the above definition only if the scale is
capable of providing that context.

If the intervals needed to connect the graph are only perceptible as justly
intoned when a particular (usually inharmonic) timbre (or class of timbres)
is used, then we say the scale is justly intoned for that timbre (or class
of timbres).
-------------------------------------------------------------------------

What goes for "JI scale" above also goes for "JI tuning" and "JI chord". I
won't attempt to define what a "significant number" is for a passage or
work or body of work.

--------------------------
Now for some homework. :-)
--------------------------

Q1. Why isn't the following a JI scale?

C#--G# Legend: 2---3 16
\ \
B---F# 19
\
A---E
\
G---D
\
F---C
\
Eb--Bb

i.e. The fifths shown are tuned 2:3 and the minor thirds shown are tuned
16:19.

Q2. Why is the following a JI scale?

C#--G# Legend: 2---3 0 cents
\ \
B---F# 294 cents
\
A---E
\
G---D
\
F---C
\
Eb--Bb

i.e. The fifths shown are tuned 2:3 and the minor thirds shown are tuned
294 cents wide.

------------
Subjectivity
------------

Dan Stearns and David Finnamore,

Deciding whether something is justly intoned or not, is somewhat
subjective, yes. But it's nothing like deciding whether a piece of music is
Jazz or not. It's much lower level than that, much closer to immediate
sensory perception than high level concepts like Jazz. It's more like
deciding whether a painting uses mostly saturated (pure or spectral)
colours as opposed to tints and shades. Whereas jazz vs. other musical
styles is like abstract vs. other painting styles. Saturated colours can
appear in any style of painting. JI can appear in any style of music.

Do we have to put up with a painter who claims that they have painted a
rainbow realistically despite the fact that they have used pink instead of
red, because they "intended" it to be red. Of course not. And if this is
offensive, then I'm sorry, but I'm duty bound to be offensive.

Coming to an agreement about what looks like a saturated colour and what
doesn't, is arrived at culturally or socially and we recognise that there
are experts in such things. We don't take much notice of people who are
clearly "colour blind". When we find that the expert viewers generally
agree, we can start taking measurements and developing mathematical models
and testing them by synthesising colours and getting the experts opinions
to check our models until we have a model that is good enough to satisfy
the experts. We've done that with colour. We now have an international
standard colour model that is no longer subjective.

We're some way off that with "sound-colour". Colour was easy. There were
only 3 receptors.

But rather than develop the maths to match the experience of experts (or
even amateurs) on what sounds justly intoned (or what degree of justness
something has), a ridiculously simple version of the mathematics has gone
wandering off on its own, proclaiming itself to be the arbiter of true JI.

And we wonder why folks often say "That JI stuff just sounds out of tune to
me". A lot of the time it's because it _is_ out of tune!

Regards,
-- Dave Keenan
http://dkeenan.com

On Tue, 28 Nov 2000 21:14:37 -0800, David C Keenan <D.KEENAN@UQ.NET.AU>
wrote:

>Consider a graph with a vertex for each pitch of the scale and an edge for
>each justly intoned interval, then a JI scale corresponds to a connected
>graph. A non-JI scale has two or more unconnected subgraphs.
>
>If an interval is only perceptible as justly intoned when in some harmonic
>context larger than the bare interval, such an interval will count as
>justly intoned for the purpose of the above definition only if the scale is
>capable of providing that context.

That sounds promising; it accounts for the Dream House as well as Ben
Johnston's harmonic scale. It excludes meantone scales, since you can't
cover the whole scale with only a sequence of major (1/4-comma) or minor
(1/3-comma) thirds. It also excludes "adaptive JI" systems like
Vicentino's, but those could be added as another class of "near-JI" scales
that don't fall under the main definition.

"Near-JI" then could be defined in reference to a connected graph with an
edge for each interval sufficiently near to JI (which is a fuzzy
definition, but probably would include schismatic temperament at least).

--
see my music page ---> ---<http://www.io.com/~hmiller/music/music.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:
> ----------------------------------
> Believe your ears, not the numbers
> ----------------------------------
> And we wonder why folks often say "That JI stuff just sounds out of
tune to
> me". A lot of the time it's because it _is_ out of tune!
>
> Regards,
> -- Dave Keenan

Dave,

Hello!

Just for the sake of deepening my understanding of this fascinating
thread, I would like to humbly ask the following question: "Out of
tune" relative to what (I realize you did say "A lot of the time")?
Our preconceptions of 5 limit harmony - or 12tET? Seems like this
could be just a little subjective (please take the liberty of
correction if seen fit). And given cultural conditioning, what we may
hear in the music of other cultures as "out of tune", would sound
completely correct within the given cultural context (e.g. Gamelan).
Probably the best example of this to me would be the acceptance of
12tET 3rds by our Western culture - and our trained ears certainly
hear this as out of tune, while those unaware of the audible
difference between a 300 and 316 cents minor 3rd, or a 400 and 386
cents major third, think they are hearing a sweetly tuned interval -
likely something that wouldn't even be perceived or questioned.

Thanks,

Jacky Ligon

P.S. Now I must recite my "context" mantra again. : )

Jacky wrote,

>--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:
>> ----------------------------------
>> Believe your ears, not the numbers
>> ----------------------------------
>> And we wonder why folks often say "That JI stuff just sounds out of
>tune to
>> me". A lot of the time it's because it _is_ out of tune!
>>
>> Regards,
>> -- Dave Keenan

>Dave,

>Hello!

>Just for the sake of deepening my understanding of this fascinating
>thread, I would like to humbly ask the following question: "Out of
>tune" relative to what (I realize you did say "A lot of the time")?

Dave's criterion is that if you can adjust the tuning of the interval a
small amount and improve its consonance and/or eliminate beating, the
original interval is out of tune. In other words, he means "out of tune"
relative to _his_ definition of JI (which seems to best correspond to the
dictionary definition). So it's strictly a harmonic criterion.

Melodically, however, even this definition of JI can sound out of tune.

>And given cultural conditioning, what we may
>hear in the music of other cultures as "out of tune", would sound
>completely correct within the given cultural context (e.g. Gamelan).
>Probably the best example of this to me would be the acceptance of
>12tET 3rds by our Western culture - and our trained ears certainly
>hear this as out of tune, while those unaware of the audible
>difference between a 300 and 316 cents minor 3rd, or a 400 and 386
>cents major third, think they are hearing a sweetly tuned interval -
>likely something that wouldn't even be perceived or questioned.

I know that Dave would wholeheartedly agree with you there.

John deLaubenfels wrote:

>In the discussion between Monz, Dave K., and others, I haven't seen one
>very large aspect of tuning addressed: if an interval targets a
>particular low integer ratio, yet is allowed to deviate somewhat from it
>for purposes of horizontal (melodic) stability, is it still "JI"? How
>much deviation is allowed? Zero point zero zero cents?

Thanks John,

I was avoiding that for as long as I could. One day at a time sweet jesus.

Many tunings (both static and dynamic) try hard to be JI while compromising
for various reasons. Merely trying hard isn't good enough. But then tunings
which are intended to be strictly JI are used on instruments (even
electronic ones) whose resolution and stability is such that +-0.5c
deviations are common.

I propose that for now, in ordinary circumstances, we accept deviations of
+-0.5 cents from strict harmonic coincidence as still being JI. This would
be convenient since it would let us round all our cents values to the
nearest whole number of cents.

------------------
Jacky Ligon wrote:

> Just for the sake of deepening my understanding of this fascinating
> thread, I would like to humbly ask the following question: "Out of
> tune" relative to what (I realize you did say "A lot of the time")?
> Our preconceptions of 5 limit harmony - or 12tET? Seems like this
> could be just a little subjective (please take the liberty of
> correction if seen fit).

Hi Jacky,

What Paul attributed to me is correct.

You'd have to subject me to a long time of some particularly hideous
torture* before I'd say that 12-tET was "in tune". (* like deprive me of my
computer for a week)

But you're right. If we relied only on a description of just intervals as
being "in tune", it would be highly subjective. Here's how we avoid that.

We define it in terms of what you can _do_ to experience it.

--------------------------------------------------------------------------
Injunctive definition of "JI interval"

Take a sustained harmonic timbre such as human voice or strings (or almost
any timbre if it's played loud enough). While sounding one tone at constant
pitch, very slowly sweep another tone up from that. Some times you hear
"wah wah". Notice when the "wah wah"s slow down and eventually stop before
speeding up again. Those sharply defined points where the "wah wah"s stop
are the "just intervals", "justly intoned intervals", "JI intervals", "pure
intervals".
--------------------------------------------------------------------------

Any definition of JI that isn't based on an injunction very like the one
above, is breaking with its historical and musical roots.

-----------
Dear Margo,

It is clear that we both recognise the same distinctions. We only disagree
about what words to use for them.

I feel that the musical distinction between pure and impure intervals is
far too important to leave to the vagaries of a capital letter,
particularly when people already use "Just" versus "just" for other
purposes unrelated to your proposal.

And we have a well-respected dictionary that claims a definition dating
from 1811 establishing "just" as synonomous with "harmonically pure".

As Joe Monzo pointed out, we already have a word for the mathematical
property of being tuned in a whole number ratio, "rational".

For "an interval that is not necessarily pure but arises in a just scale" I
suggest simply "JI-scale interval". The hyphen is important to avoid the
interpretation "JI scale-interval".

For "an interval that is definitely not pure, in a just scale" I suggest
"non-just JI-scale interval".

----------------------------
Snappy-but-rough definitions
----------------------------

It's natural to want a snappy definition of JI, even if it's only
approximate. But "tuning by whole number ratios" isn't just rough, it's
just wrong.

I'm pleased that John Chalmers has repudiated "tuning by whole number
ratios". Here's where David Doty makes the same mistake John did.

Short version
http://www.dnai.com/~jinetwk/whatisji.html
Long version (first chapter of David Doty's Just Intonation Primer).
http://www.dnai.com/~jinetwk/primer2.html

"Tuning by small whole number ratios" is much better, but as Dave Beardsley
pointed out, "small" can mean anything from 10 to 200 depending heavily on
harmonic context.

How about:
-------------------------
JI is tuning by harmonics
-------------------------

Regards,
-- Dave Keenan
-- Dave Keenan
http://dkeenan.com

[I wrote:]
>>In the discussion between Monz, Dave K., and others, I haven't seen
>>one very large aspect of tuning addressed: if an interval targets a
>>particular low integer ratio, yet is allowed to deviate somewhat from
>>it for purposes of horizontal (melodic) stability, is it still "JI"?
>>How much deviation is allowed? Zero point zero zero cents?

[Dave Keenan:]
>Thanks John,
>I was avoiding that for as long as I could. One day at a time sweet
>jesus.

Sorry, Dave! ;-)

>Many tunings (both static and dynamic) try hard to be JI while
>compromising for various reasons. Merely trying hard isn't good
>enough.

Agreed.

>But then tunings which are intended to be strictly JI are used on
>instruments (even electronic ones) whose resolution and stability is
>such that +-0.5c deviations are common.

I would guess that even a very fine acoustic guitar (to pick an
instrument used by many list members) would have deviations much greater
than this - does anyone know if measurements have been made?

>I propose that for now, in ordinary circumstances, we accept deviations
>of +-0.5 cents from strict harmonic coincidence as still being JI. This
>would be convenient since it would let us round all our cents values to
>the nearest whole number of cents.

That's very strict, but for all I know there may be list members who
protest that it's not strict enough. So much depends upon whether there
are long sustained dyads, and whether the composer is willing to accept
very slow beating as still representing JI.

The range you suggest is too small for any significant horizontal
(melodic) retuning relief, so the spring coefficients I choose in my
work, which DO allow for significant horizontal relief, take me well
outside "true JI" by this definition - I routinely accept vertical
deviations of several cents. "Quasi-JI", then (as suggested by Paul E),
would be the correct term for this, do you agree, Dave?

JdL

> >But then tunings which are intended to be strictly JI are used on
> >instruments (even electronic ones) whose resolution and stability
is
> >such that +-0.5c deviations are common.
>
> I would guess that even a very fine acoustic guitar (to pick an
> instrument used by many list members) would have deviations much
greater
> than this - does anyone know if measurements have been made?

I would guess that too. Apart from stablity of string tension and
accuracy of fret placement, I should think a very small change in
finger pressure between frets can cause many cents change.

> >I propose that for now, in ordinary circumstances, we accept
deviations
> >of +-0.5 cents from strict harmonic coincidence as still being JI.
This
> >would be convenient since it would let us round all our cents
values to
> >the nearest whole number of cents.
>
> That's very strict, but for all I know there may be list members who
> protest that it's not strict enough.

I hope not, but I'm listening.

With the +-0.5 cent allowance, the smallest ETs that qualify as just
at various odd limits are:

Odd-limit Smallest just ET
3 53
5 118
7 171
9 171
11 342

> So much depends upon whether
there
> are long sustained dyads, and whether the composer is willing to
accept
> very slow beating as still representing JI.

I personally think that very slow beating is more interesting to
listen to, and in any case hard to avoid.

> The range you suggest is too small for any significant horizontal
> (melodic) retuning relief, so the spring coefficients I choose in my
> work, which DO allow for significant horizontal relief, take me well
> outside "true JI" by this definition - I routinely accept vertical
> deviations of several cents. "Quasi-JI", then (as suggested by Paul
E),
> would be the correct term for this, do you agree, Dave?

Yes. I'd say 1/4-comma meantone is on the upper limit of 5-(odd)limit
quasi-just, with its 5.4 cent errors in the fifths and fourths. But
quasi-just should probably be defined somehow in terms of beat rate,
so that more complex ratios aren't allowed to deviate as much.
Inversely proportional to Lamothe complexity (a*b)? Maybe plain old
"just" should be defined in that way too.

Regards,
-- Dave Keenan

[I wrote:]
>>The range you suggest is too small for any significant horizontal
>>(melodic) retuning relief, so the spring coefficients I choose in my
>>work, which DO allow for significant horizontal relief, take me well
>>outside "true JI" by this definition - I routinely accept vertical
>>deviations of several cents. "Quasi-JI", then (as suggested by Paul
>>E), would be the correct term for this, do you agree, Dave?

[Dave Keenan:]
>Yes. I'd say 1/4-comma meantone is on the upper limit of 5-(odd)limit
>quasi-just, with its 5.4 cent errors in the fifths and fourths. But
>quasi-just should probably be defined somehow in terms of beat rate,
>so that more complex ratios aren't allowed to deviate as much.
>Inversely proportional to Lamothe complexity (a*b)? Maybe plain old
>"just" should be defined in that way too.

Hmmm: I understand the math that says that allowable deviation in cents
should narrow as the interval gets more complex, yet to my ear five
cents deviation in a fifth is more painful than five cents in a major
third. I think this may be because the amplitude of the partials that
meet (or beat) in a 3:2 is usually higher than the amplitude of higher
partials that meet (or beat) in a more complex interval.

I ask other list members to share their impressions as to the
sensitivity of mistuning of simpler vs. more complex intervals. And
let's not forget a 2:1 if we're addressing the question. Or even 1:1,
for that matter! (though, as a practical matter, 2:1 and 1:1 can be
properly tuned without difficulty).

JdL

John deLaubenfels wrote,

>Hmmm: I understand the math that says that allowable deviation in cents
>should narrow as the interval gets more complex, yet to my ear five
>cents deviation in a fifth is more painful than five cents in a major
>third.

"The math" actually agrees with your ear, John, if "the math" used is
harmonic entropy or a consideration of beat rates that weights them by the
typical amplitudes of the partials involved.

>I think this may be because the amplitude of the partials that
>meet (or beat) in a 3:2 is usually higher than the amplitude of higher
>partials that meet (or beat) in a more complex interval.

Right . . .

However, John, things get more complicated when you're including these
various intervals in the same chord . . .

Jacky Ligon wrote,

>> I would guess that even a very fine acoustic guitar (to pick an
>> instrument used by many list members) would have deviations much
greater
>> than this - does anyone know if measurements have been made?

Dave Keenan wrote,

>I would guess that too. Apart from stablity of string tension and
>accuracy of fret placement, I should think a very small change in
>finger pressure between frets can cause many cents change.

It is almost impossible to change the pitch by even one cent by even greatly
changing finger pressure perpendicular to the fingerboard between frets,
except on an electric guitar with super-light strings. The usual methods of
obtaining vibrato are by applying pressure in one of the other two
directions, either parallel to the frets or parallel to the string. While
inexperienced players will often inadvertantly effect some of these motions
when fingering difficult chords, etc., a good guitarist will be able to keep
these to a minimum (say, two or three cents) when desired -- especially on
the acoustic guitar with its much greater string tension.

However, Jacky may be right in that guitars fretted without corrections for
string height and with a straight nut can have errors often as large as 10
cents on the lowest frets. Look at Dante Rosati's guitar, on which he placed
frets separately for each string by ear, for confirmation:

http://users.rcn.com/dante.interport/justguitar.html.

[I wrote:]
>>Hmmm: I understand the math that says that allowable deviation in
>>cents should narrow as the interval gets more complex, yet to my ear
>>five cents deviation in a fifth is more painful than five cents in a
>>major third.

[Paul E:]
>"The math" actually agrees with your ear, John, if "the math" used is
>harmonic entropy or a consideration of beat rates that weights them by
>the typical amplitudes of the partials involved.

[JdL:]
>>I think this may be because the amplitude of the partials that
>>meet (or beat) in a 3:2 is usually higher than the amplitude of higher
>>partials that meet (or beat) in a more complex interval.

[Paul:]
Right . . .

>However, John, things get more complicated when you're including these
>various intervals in the same chord . . .

Yes. But Dave Keenan is making the case (unless I'm misunderstanding)
that more complex dyads have less tuning tolerance than less complex
dyads. It sounds like you're agreeing with me (?) that this is not the
case.

It seems we are all struggling with the question of how to understand,
and calculate, the cumulative effects of tuning, and mistuning, chords
larger than dyads.

JdL

[Paul E wrote:]
>Jacky Ligon wrote,
>>>I would guess that even a very fine acoustic guitar (to pick an
>>>instrument used by many list members) would have deviations much
>>>greater than this - does anyone know if measurements have been made?

Actually, that was me, Paul.

JdL

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:

> Yes. But Dave Keenan is making the case (unless I'm misunderstanding)
> that more complex dyads have less tuning tolerance than less complex
> dyads. It sounds like you're agreeing with me (?) that this is not the
> case.

Given the uncertainties involved when dyads are involved in larger chords, as
well as the issue that more complex dyads may resemble a different ratio
altogether if mistuned (all things I've discussed at length), if I had to use a
generic rule it would be a fixed tuning tolerance across all dyads.
>
> It seems we are all struggling with the question of how to understand,
> and calculate, the cumulative effects of tuning, and mistuning, chords
> larger than dyads.
>
There, as we've seen, the issue gets quite complex -- recently we discussed
how a "wolf" fifth can sound altogether pleasant when part of a major triad
where each of the thirds is only half as out-of-tune as the fifth. I think triadic
harmonic entropy may address a lot of these issues -- I just got the new
version of Matlab so I might start up some calculations next week and see
how long they take.

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

http://www.egroups.com/message/tuning/16147

I think triadic harmonic entropy may address a lot of these issues --
I just got the new version of Matlab so I might start up some
calculations next week and see how long they take.

OK... Looking forward to more "pizza" being served!
________ ___ __ _
Joseph Pehrson

Carl Lumma wrote:

>> One issue which has caused much confusion in this thread needs to
>> be addressed -- the phrase "just intonation" is sometimes being
>> used to mean a _scale_, and other times, a _sonority_.

I thought I addressed it quite thoroughly. Just intonation is either a
process, or a property that something may or may not posess (or may posess
to some degree). That something may certainly be either a scale or a
sonority. I see no confusion. A _scale_ is only justly intoned if it allows
enough _sonorities_ which are justly intoned.

>> The definition quoted from _Tetrachord_ is one of the former type,

Here it is again:

From John Chalmers, Divisions of the Tetrachord (as quoted in Joe Monzo's
dictionary):
-------------------------------------------------------------------------
Just intonation

Any tuning system which exclusively employs intervals defined by ratios of
integers may be called Just Intonation, though some authors restrict it to
systems whose intervals are derived from the first six overtones, 1, 2, 3,
4, 5, and 6. Such systems are often termed "Five Limit" or "Senary" systems
after Zarlino's "senario" (Partch, 1949, 1974, 1979). The most common
example of such a system is the tuning of the Major Mode as 1/1 9/8 5/4 4/3
3/2 5/3 15/8 and 2/1.

Just Intonation is contrasted to Equal Temperament and Unequal Temperaments
such as Meantone which combine rational with irrational intervals.
-------------------------------------------------------------------------

The part of that definition that I disagree with (and John Chalmers has
already repudiated (correct me if I'm wrong John)) is:

"Any tuning system which exclusively employs intervals defined by ratios of
integers may be called Just Intonation, ..."

I see what you mean about the grammar. I would have written "may be called
Justly Intoned". But the abbreviation JI works either way and this seems a
minor grammatical quibble compared to the major issue of whether JI is
something you can _hear_, or whether it really only depends on whether its
designer used integer math or not.

>> and one which even refers to a particular scale. As John pointed
>> out, there is excellent support for this definition

Which definition? _Any_ integer ratios? First 6 overtones? 5-odd-limit
ratios? The particular scale?

>>-- in fact, it
>> may be the most accurate definition outside of this community.

Since you say this, I guess you mean either 5-limit JI or the particular
scale 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1.

>> In order to generalize this definition for any scale, well... it's
>> a damn challenge.

So what's wrong with my attempt?

>> And the fruits of such a labor are probably non-
>> existent.

I totally disagree.

>> Which is why I suggested we forget the idea, and stick
>> to definitions of the latter type.

So what's your favourite definition of a JI (justly intoned) sonority?

I get the feeling that there's a lotta people out there that agree with me
when I say "believe your ears, not the numbers", but are afraid to come
right out and say so. I've so far had two people email me privately to
thank me for clarifying what JI actually is.

-----------------
Is barbershop JI?
-----------------

I'd like to point out that if JI is simply "any integer ratios" then it is
impossible to tell whether barbershop singing is JI or not, but we can
certainly tell that it is not 19-limit JI. It might be 1,000,001-limit.

To me barbershop epitomises JI (and in particular 19-limit JI), and there
aint no integer math involved.

----------------
Homework answers
----------------

Here are the answers to my "homework" questions.

Q1. Why isn't the following a JI scale?

C#--G# Legend: 2---3 16
\ \
B---F# 19
\
A---E
\
G---D
\
F---C
\
Eb--Bb

i.e. The fifths shown are tuned 2:3 and the minor thirds shown are tuned
16:19.

Answer 1.

A bare 16:19 (or 8:19 or 4:19 etc.) interval cannot be heard as justly
intoned (with ordinary timbres). That is, no beat cancellation or
clarification can be perceived when it is retuned after being detuned. And
the scale does not provide any larger context for the 16:19 (such as
16:19:24) which might be perceived as justly intoned. Therefore the pitches
are not fully connected by just intervals. Therefore it is not a just scale.

Q2. Why is the following a JI scale?

C#--G# Legend: 2---3 0 cents
\ \
B---F# 294 cents
\
A---E
\
G---D
\
F---C
\
Eb--Bb

i.e. The fifths shown are tuned 2:3 and the minor thirds shown are tuned
294 cents wide.

Answer 2.

This one was tricky. 294 cents is no more justly intoned than is 16:19,
however this value of minor third happens to result in other fifths which
are within 0.2 cents of 2:3 (Bb:F, C:G, D:A, E:B, F#:C#). These fifths will
definitely be perceived as justly intoned and so the pitches of the scale
are completely connected as a chain of fifths. Therefore this is a just
scale, in fact a Pythagorean scale.

Regards,
-- Dave Keenan
http://dkeenan.com

What follows is a slightly edited version of a private email from myself to
Margo Schulter. Margo will follow it with an edited version of her reply.
We particularly ask for comment from Jacky Ligon.
-----------------------------------------------------------------------

Dear Margo,

You wrote:
>The dialogue we have seen here from many able contributors has
>convinced me, at least, that the term "JI" has such different
>associations and premises for different people that the best we can do
>is to acknowledge and indeed cherish this diversity.

There is really only one major point at issue here. Is JI fundamentally
(a) a mathematical property (which turns out to be inaudible and
immeasurable) or
(b) an audible property (which for convenience we model mathematically)?

One should certainly cherish diversity of musical forms. Such diversity
will remain no matter what we call them. But I don't feel inclined to
cherish diversity in the meaning of a word or phrase. Such diversity only
leads to misunderstanding and thereby suffering.

Rather than give up a definition of JI as "a system of tuning based on
integer ratios", you have said that the Hammond organ tuning is JI, despite
the fact that it cannot be distinguished from 12-tET by listening to it.

I have a synthesizer which is entirely digital, i.e. its frequencies are
all derived from a single master oscillator purely by integer arithmetic.
Its semitones approximate the 12th root of 2 far better even than the
Hammond organ, and yet, by your definition it is tuned in just intonation.

We choose definitions to serve us. I feel very badly served, and so I
suspect does Jacky Ligon, by a definition of JI that leads to inaudible
distinctions such as these.

I'd also like to point out that, by your definition, barbershop singing is
not JI (or at least we cannot tell whether it is or not). As you would well
be aware, there are no mathematical calculations involved in barbershop
singing, let alone integer ratios. It is done entirely by auditory
feedback. The singers listen to themselves to ensure that their chords are
justly intoned (by my definition).

You might say "but we can measure them".

No measurement can ever be so precise as to distinguish between a rational
and an irrational number. But even a moderately precise measurement would
show that they are certainly not using integer ratios where the integers
are limited to 19 or less. One would probably need to go beyond 3 digits to
represent the measurement results as integer ratios within the resolution
of the instrument, and they would be different every time. The more precise
the instrument, the larger the integers you would need to use.

I maintain that barbershop singing _epitomises_ JI and can accurately be
described as 19-limit JI.

If you agree that Barbershop is JI, as most people do, how do you square
this with your definition.

What is it that you fear, if we were to adopt a definition of JI based
primarily on sound, not numbers. Are you afraid of losing something
valuable. If so, please try to explain to me what it is.

In friendship,
-- Dave Keenan
http://dkeenan.com

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:

http://www.egroups.com/message/tuning/16160

Thanks to Dave Keenan for the interesting (and continuing) discussion
of Just Intonation definitions and, particularly, for the puzzles.

Paul Erlich has done "puzzles" on this list as well and, although I
know there are people on the list who really dislike such practices,
there are others, like myself (probably newer to the topics) that
really like them. (Whether we get them right off or not!)

Thanks again!

_________ ___ __ _
Joseph Pehrson

>>One issue which has caused much confusion in this thread needs to
>>be addressed -- the phrase "just intonation" is sometimes being
>>used to mean a _scale_, and other times, a _sonority_.
>
>I thought I addressed it quite thoroughly. Just intonation is either a
>process, or a property that something may or may not posess (or may
>posess to some degree). That something may certainly be either a scale
>or a sonority. I see no confusion.

Perhaps you did make the distinction -- forgive me. But I did see
confusion on this issue among other posts in this thread.

>A _scale_ is only justly intoned if it allows enough _sonorities_ which
>are justly intoned.

And which ones are those? How much is enough? And if I use the scale
only for dissonance, is it still a "just" tuning?

>The part of that definition that I disagree with (and John Chalmers has
>already repudiated (correct me if I'm wrong John)) is:
>
>"Any tuning system which exclusively employs intervals defined by ratios
>of integers may be called Just Intonation, ..."
//
>the major issue of whether JI is something you can _hear_, or whether it
>really only depends on whether its designer used integer math or not.

I would prefer to make it something one can hear, but for scales, it's
just too hard a problem. Methods don't nearly exist to solve it, and
it seems unreasonable to require such analysis for the use of such an
established phrase.

>>and one which even refers to a particular scale. As John pointed
>>out, there is excellent support for this definition
>
>Which definition? _Any_ integer ratios?

I meant the intense diatonic. But there's also excellent support for
the integer math definition, especially within our community. Lots of
folks use the term that way. Definition is a descriptive endeavor,
remember. Because I agree with you, I do suggest dropping the use of
"just intonation" in reference to scales, but that is as far as I will go.

>>in fact, it may be the most accurate definition outside of this community.
>
>Since you say this, I guess you mean either 5-limit JI or the particular
>scale 1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1.

Correct.

>>And the fruits of such a labor are probably non-existent.
>
>I totally disagree.

People will still use their scales, and I doubt that any secrets of
music are hiding in the distinction between a just scale (as you define
it) and not. There are secrets to discover solving the tolerance
problem, and we are not yet there. But even when we arrive, another
very difficult, and I will claim fruitless, level is needed to apply it
to scales and yield your definition.

>>Which is why I suggested we forget the idea, and stick to definitions
>>of the latter type.
>
>So what's your favourite definition of a JI (justly intoned) sonority?

I very much like:

"A group of pitches is said to be justly intoned when beating cannot
be heard."

A potential weakness of this definition is that it allows a single
complex tone (IOW, a timbre) to be considered JI. However, I actually
consider this a strength. Can anyone guess why?

>I get the feeling that there's a lotta people out there that agree with
>me when I say "believe your ears, not the numbers", but are afraid to
>come right out and say so. I've so far had two people email me privately
>to thank me for clarifying what JI actually is.

Dave, that's unfair -- you know I agree with you.

>I'd like to point out that if JI is simply "any integer ratios" then it
>is impossible to tell whether barbershop singing is JI or not, but we
>can certainly tell that it is not 19-limit JI. It might be
>1,000,001-limit.

That depends on the resolution, which we must assume regardless of the
definition.

-Carl

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:
> What follows is a slightly edited version of a private email from
myself to
> Margo Schulter. Margo will follow it with an edited version of her
reply.
> We particularly ask for comment from Jacky Ligon.
> --------------------------------------------------------------------
---
>
> Dear Margo,
>
> You wrote:
> >The dialogue we have seen here from many able contributors has
> >convinced me, at least, that the term "JI" has such different
> >associations and premises for different people that the best we
can do
> >is to acknowledge and indeed cherish this diversity.
>
> There is really only one major point at issue here. Is JI
fundamentally
> (a) a mathematical property (which turns out to be inaudible and
> immeasurable) or
> (b) an audible property (which for convenience we model
mathematically)?

Dave,

Hello!

Perhaps a little of both. Is sub atomic matter "particles"
or "waves"? Depends on how you choose to look at it.

>
> Rather than give up a definition of JI as "a system of tuning based
on
> integer ratios", you have said that the Hammond organ tuning is JI,
despite
> the fact that it cannot be distinguished from 12-tET by listening
to it.
>
> I have a synthesizer which is entirely digital, i.e. its
frequencies are
> all derived from a single master oscillator purely by integer
arithmetic.
> Its semitones approximate the 12th root of 2 far better even than
the
> Hammond organ, and yet, by your definition it is tuned in just
intonation.

The most interesting fact here for me, is that even such a perverse
arrangement of pitches as this is found in the spectrum of integer
ratios. This is using the ratios to approximate the temperament, and
no knowing JI composer would deliberately seek this out as an example
of a JI tuning. It's a tough call, but I think the goal with the
tuning of these tempered instruments is quite a different concern
than that of tuning by ratios in the normal JI sense - which would
likely yield a scale of uneven steps.

>
> We choose definitions to serve us. I feel very badly served, and so
I
> suspect does Jacky Ligon, by a definition of JI that leads to
inaudible
> distinctions such as these.

Since this quandary of terminology is a concern, as far as detailing
compositional notes on upcoming CD releases, I would like to put
forth a few questions on this topic by way of musical scenario.

It is an important part of my compositional process, to transform the
tunings of individual instruments, or groups of timbres over the
course of a composition. It is common for me to introduce more
complexity as the composition evolves - which may include starting
with simple JI intervals, and evolving to more complex higher primes.
Suppose at the beginning of a piece, I use a simple mode capable of
playing justly intoned "beatless" intervals, such as:

Ratio Cents Value
1/1 0
9/8 203.9100017
5/4 386.3137139
4/3 498.0449991
3/2 701.9550009
8/5 813.6862861
16/9 996.0899983
2/1 1200

And imaging this mode is used for the first 20-30 seconds of the
piece, playing on perhaps flutes and strings, when I might introduce:

Ratio Cents Value
1/1 0
21/20 84.46719347
21/19 173.2678912
28/23 340.5515592
26/21 369.7467544
30/23 459.9943675
40/29 556.7365197
29/20 643.2634803
23/15 740.0056325
21/13 830.2532456
23/14 859.4484408
38/21 1026.732109
40/21 1115.532807
2/1 1200

Playing on a Santoor. After about 10 seconds I might return to the
simple mode again for a reiteration of it's "mood" for another 15
seconds, when I might introduce completely new timbres and a new
tuning, such as:

0: 1/1 0.000
1: 206720/180879 231.184
2: 221792/180879 353.019
3: 1248926/904395 558.796
4: 1378112/904395 729.202
5: 103360/60293 933.139
6: 1754956/904395 1147.692
7: 1798022/904395 1189.663

A spectrum mode for tuned Burmese Nipple Gongs. This tuning sounds
completely correct and concordant playing on this timbre - the fifth
sounds in tune, although I'm not sure one would call it "beatless"
since metal instruments can beat within the timbre, but in this case
the tuning matches the timbre.

At this point of the compositional scenario, I make the flutes heard
in the opening change their tuning to that of the Burmese nipple
gongs, for a kind lovely tension, followed immediately by a
sectional "resolve" where the flutes/strings again change back to the
original simple major mode and are joined by the Santoor.

Now the questions:

1. Is my composition in Just Intonation?

2. Or is only a part of it in JI?

3. At what point (prime - odd - whatever) do I cross the barrier of
complexity and disqualify my composition from the label of "JI"?

4. How do I reconcile the fact that my piece has a range of simple to
complex integer ratios, sometimes "justly intoning" beatless
intervals, and alternately injecting tuning shifts to more complex
structures, which might not fit the text book classification of JI?

5. What the heck do you call music that morphs it's tunings in this
manner, if not JI?

As I see it, there has been for many, a general evolution of the
meaning of the term JI; one that has perhaps broadened to accommodate
our contemporary sensibilities, and desire for more complex
structures. And I see your point about the generally accepted
meaning. The difficultly for me personally, is that if a composer is
using as the foundation of their harmonic vocabulary "justly intoned"
intervals, yet also includes ratios which do not fit the beatless
criteria as a part of the compositional "flavor", then perhaps we can
allow this stretch of terminology.

I had never questioned that what I was doing was JI or not, until my
arrival on the Tuning List - I always assumed it was JI, because of
the above rationale. But even considering this, I hardly feel it's
all that important to quibble about, since my main goal is to use a
broad range of simple to complex ratios in my compositions,
regardless of labels. In the end it's "Music" that I'm concerned
with. The most important label of all!

>
> I maintain that barbershop singing _epitomizes_ JI and can
accurately be
> described as 19-limit JI.
>
> If you agree that Barbershop is JI, as most people do, how do you
square
> this with your definition.

Ah, so it's "19"!? Well that leaves me a wanderer in the wilderness
of the ones without name! It is my direct experience though, that
singers can be trained to sing higher than 19 prime with remarkable
precision. In fact this will soon become a priority area of musical
focus - the results are so compelling!

> What is it that you fear, if we were to adopt a definition of JI
based
> primarily on sound, not numbers.

Perhaps it is appropriate to not discount either. There's no resource
I'm willing to reject.

> Are you afraid of losing something
> valuable. If so, please try to explain to me what it is.

My fear is that we might lose the flexibility to expand this
definition to include the broader meaning that it has assumed for
many. Who wants to be the person to tell Kraig Grady or La Monte
Young - or anyone using high prime ratios, that what they call Just
Intonation, has been ruled incorrect by our friendly forum? Surely a
consensus here would be meaningless without the input from these
masters, who have helped to broaden the meaning of JI.

Thanks Dave, for the wonderfully penetrating questions into this
topic,

Jacky Ligon

P.S. Kyle Gann, is it not correct that you define JI as tuning by
integer ratios?

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:
> >A _scale_ is only justly intoned if it allows enough _sonorities_
which
> >are justly intoned.
>
> And which ones are those? How much is enough?

It seems you missed http://www.egroups.com/message/tuning/16004

> And if I use the scale
> only for dissonance, is it still a "just" tuning?

Sure.

> I would prefer to make it something one can hear, but for scales,
it's
> just too hard a problem. Methods don't nearly exist to solve it,
and
> it seems unreasonable to require such analysis for the use of such
an
> established phrase.

I guess my definition of JI, for other than intervals, isn't something you
can necessarily hear by auditioning it in any particular piece(s) of music.
But you can definitely "hear" it if you are allowed to listen to every
interval, while knowing what pitches they contain.

So what you think of the "no remaining freedom to justly intone" idea, as
referenced above.

> >So what's your favourite definition of a JI (justly intoned)
sonority?
>
> I very much like:
>
> "A group of pitches is said to be justly intoned when beating cannot
> be heard."

Excellent. Except that beats will also not be heard on maximally rough
intervals and yet I wouldn't call them justly intoned. That is why I say
that the test also needs to check whether you _do_ hear beats when you
mistune them slightly (say up to 5 cents).

I also say that a justly intoned sonority is allowed to have beats. So long
as it is impossible to eliminate the beats due to one pair of notes without
introducing beats due to another pair. The entire sonority must be
connected by just intervals, but not every interval needs to be just.
Exactly the same as the definition of a JI scale.

i.e a just chord, scale or tuning must have a "spanning tree" of just
intervals.

To me, this definition works like magic and keeps most people happy
(notably, not Monz).

Monz, I don't think there is any shame in admitting that you tempered one
chord. It doesn't mean your piece is not JI. You will not be excommunicated
from the church of JI. :-) Margo coined the delightful phrase "tempering by
ratios" for what you did with the 64:75:96.

> A potential weakness of this definition is that it allows a single
> complex tone (IOW, a timbre) to be considered JI. However, I
actually
> consider this a strength. Can anyone guess why?

Not me.

> Dave, that's unfair -- you know I agree with you.

Sorry Carl.

Regards,
-- Dave Keenan
http://dkeenan.com

In case anyone's still looking for Margo's reply, it's here:
http://www.egroups.com/message/tuning/16183

Correction:

For some dumb reason, I said barbershop was 19-limit JI. I think it's
typically only 17-limit, and it needs a tetrad or pentad to get to 17.
Namely the diminished seventh chord tuned 10:12:14:17, or the dominant flat
ninth tuned 8:10:12:14:17.

-- Dave Keenan
http://dkeenan.com

>>And which ones are those? How much is enough?
>
>It seems you missed http://www.egroups.com/message/tuning/16004

So I did. You must mean:

>A justly intoned scale is is one where every pitch can be reached from
>every other pitch by some chain of justly intoned intervals.

Sorry, Dave, but this sounds rather arbitrary and a little too strong.

>If the intervals needed to connect the graph are only perceptible as
>justly intoned when a particular (usually inharmonic) timbre (or class
>of timbres) is used, then we say the scale is justly intoned for that
>timbre (or class of timbres).

We always need this, so I think any good definition should assume it
from the beginning (IOW, it should not assume harmonic timbres). See
below.

>So what you think of the "no remaining freedom to justly intone" idea,
>as referenced above.

Sorry, don't see it.

>>>So what's your favourite definition of a JI (justly intoned) sonority?
>>
>>I very much like:
>>
>>"A group of pitches is said to be justly intoned when beating cannot
>>be heard."
>
>Excellent. Except that beats will also not be heard on maximally rough
>intervals and yet I wouldn't call them justly intoned.

Maybe you can't count the beats, but surely you're hearing the beating
as roughness!

>I also say that a justly intoned sonority is allowed to have beats. So
>long as it is impossible to eliminate the beats due to one pair of notes
>without introducing beats due to another pair. The entire sonority must
>be connected by just intervals, but not every interval needs to be just.

Doesn't this include chords like (1/1: 5/4: 25/16 : 7/4)?

>>A potential weakness of this definition is that it allows a single
>>complex tone (IOW, a timbre) to be considered JI. However, I
>>actually consider this a strength. Can anyone guess why?
>
>Not me.

Precisely because I consider music to be a sort of meta-timbre. If
we try to restrict "JI" to the meta (the building blocks must already
be complex tones), we wind up having to explain that the timbres
must match the meta-tuning anyway!

-Carl

--- In tuning@egroups.com, David C Keenan <D.KEENAN@U...> wrote:

> http://www.egroups.com/message/tuning/16186
>
> I also say that a justly intoned sonority is allowed to have
> beats. So long as it is impossible to eliminate the beats due
> to one pair of notes without introducing beats due to another
> pair. The entire sonority must be connected by just intervals,
> but not every interval needs to be just.
> Exactly the same as the definition of a JI scale.
>
> i.e a just chord, scale or tuning must have a "spanning tree"
> of just intervals.
>
> To me, this definition works like magic and keeps most people happy
> (notably, not Monz).
>
> Monz, I don't think there is any shame in admitting that you
> tempered one chord. It doesn't mean your piece is not JI. You
> will not be excommunicated from the church of JI. :-) Margo
> coined the delightful phrase "tempering by ratios" for what
> you did with the 64:75:96.

I've been trying to keep a relatively low profile thruout this
discussion, partly because I really haven't had enough time to
consider deeply all the different opinions, and partly because I
think the definitions in my Dictionary still reflect pretty well
my own opinions on the matter. I do plan on expanding those
definitions considerably, but I want to digest all the different
opinions presented here first and try to sum them up succintly
before making the changes to my webpages.

I agree with you, Dave K., that any concrete definition of
'just-intonation' or 'JI' (or any of the variants with or
without capitals, hyphens, abbreviations, etc.) should be
based primarily on *auditory* considerations - especially, but
not limited to, beatlessness - (how's that for attorney-ese?),
as well as on the ability of relatively low-integer rational
mathematics, based perhaps mainly on the harmonic-series-as-
archtype, to model these effects.

This is precisely why I put qualifications into my Dictionary
definitions about historical usage of the terms, limitations
to 5-limit, etc.

IMO, rational tunings using prime-factors higher than 5 are
best referred to as '*extended* just-intonation'. I'd tend to
say that even this term would not apply to tunings using
prime-factors above, say, 37 or 43, except in specific
circumstances such as La Monte Young's compositions, where
the style of performance demands that even primes above
200 be heard as harmonic ratios. My reasons for choosing
those particular primes are based on the prime-limits
encountered in so-called 'extended JI' compositions by Ben
Johnston, Ezra Sims, and myself.

But, again IMO, the terms 'just-intonation' or 'JI' without
further qualification should really mean specifically 5-limit
JI *where prime-factor 5 never has an absolute exponent limit
higher than 1 in any given sonority*. This pretty much amounts
to an integer-limit of 24 within a prime-limit of 5. I would
cite mainly historical usage of the term over the last 500 years
to defend my position on this.

To my mind, a composer like Partch, whose stipulated tuning
system was 11-limit (prime or odd), and who referred to it as
'just', represents the grey area between these attempts at
precision. But please note that Partch was also careful to
employ the term 'Monophony' to describe his specific tuning
system.

I stand by my use of the term 'rational tuning' for all rational
tunings generally, and for those above the 'extended JI' limits
(37, 43...?) specifically.

Dave K., I believe I've already admitted clearly in previous posts
that I agree with you that 'tempering' is one way of describing
what I did in _3 Plus 4_ in regard to the 64:75:96 triad. But I
also said (agreeing with David Finnamore) that 'extended reference'
came into play too, in my decision to tune that 'minor 3rd' as
64:75. Perhaps I've been a bit sloppy in describing this piece
and others on my website as 'JI' without using the qualification
'extended', but let it be said here that I still feel that the
tuning of the 1999 version of _3 Plus 4_ is best characterized
as 'extended just-intonation'.

Given my very personal ideas about religion in general, I'm
certainly not at all concerned about being kicked out of the
'Church of JI', or any other. But as the author of the online
Tuning Dictionary, I think it behooves me to engage myself
considerably in ascertaining a definition of just-intonation
with which the largest number of composers, theorists, and
- yes - mathematicians will agree, and which will enable all
of us to continue theoretical discourse which makes abundant
use of the term and still makes sense.

Given the constraints on my time these days, I humbly invite
anyone interested in this thread to submit *concise* corrections
and additions to my Dictionary entry:

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

Please post them here first so they may be subject to debate.

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

[Dave Keenan wrote...]
>For some dumb reason, I said barbershop was 19-limit JI. I think it's
>typically only 17-limit, and it needs a tetrad or pentad to get to 17.
>Namely the diminished seventh chord tuned 10:12:14:17, or the dominant flat
>ninth tuned 8:10:12:14:17.

I didn't stop you on that one, because I'm sure many minor triads are
drawn to 16:19:24. Nevertheless, I agree with this statement totally.
Bravo for pointing out that the term "17-limit" is not very useful here
either, since this music skips stuff between 9 and 17, and uses, to my
knowledge, 17 only for the 8:10:12:17, as you say.

-Carl

--- In tuning@egroups.com, Carl Lumma <CLUMMA@N...> wrote:

> http://www.egroups.com/message/tuning/16193
>
> [Dave Keenan wrote...]
> > For some dumb reason, I said barbershop was 19-limit JI.
> > I think it's typically only 17-limit, and it needs a tetrad
> > or pentad to get to 17. Namely the diminished seventh chord
> > tuned 10:12:14:17, or the dominant flat ninth tuned
> > 8:10:12:14:17.
>
> I didn't stop you on that one, because I'm sure many minor
> triads are drawn to 16:19:24. Nevertheless, I agree with this
> statement totally.
> Bravo for pointing out that the term "17-limit" is not very
> useful here either, since this music skips stuff between 9
> and 17, and uses, to my knowledge, 17 only for the 8:10:12:17,
> as you say.

I agree with this too. I'd use the description '7-/17-limit'
for barbershop, since it doesn't imply 11 or 13. Even tho
17 is only implied in the 'diminished 7th' chord, that chord
is used a *lot* in barbershop.

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

[Monz wrote:]
>IMO, rational tunings using prime-factors higher than 5 are
>best referred to as '*extended* just-intonation'. I'd tend to
>say that even this term would not apply to tunings using
>prime-factors above, say, 37 or 43, except in specific
>circumstances such as La Monte Young's compositions, where
>the style of performance demands that even primes above
>200 be heard as harmonic ratios.

I think that's a bad division. While slicing off everything above 5 may
have historical rationalization, I believe that aural measurements would
extend JI at least to 7 (and probably most of the way thru the next
octave) before calling it "extended".

Putting 7 in the same group as 37 or 43, or the chord 64:75:96, seems
to me disconnected from what we can hear. A limit closer to what is
sung, and heard, in barbershop music makes more sense to me.

JdL

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:

> http://www.egroups.com/message/tuning/16197

> [Monz wrote:]
> > IMO, rational tunings using prime-factors higher than 5 are
> > best referred to as '*extended* just-intonation'. I'd tend to
> > say that even this term would not apply to tunings using
> > prime-factors above, say, 37 or 43, except in specific
> > circumstances such as La Monte Young's compositions, where
> > the style of performance demands that even primes above
> > 200 be heard as harmonic ratios.
>
> I think that's a bad division. While slicing off everything
> above 5 may have historical rationalization, I believe that
> aural measurements would extend JI at least to 7 (and probably
> most of the way thru the next octave) before calling it
> "extended".
>
> Putting 7 in the same group as 37 or 43, or the chord 64:75:96,
> seems to me disconnected from what we can hear. A limit closer
> to what is sung, and heard, in barbershop music makes more sense
> to me.

Well... I see your point, which is precisely why I mentioned
Partch's 11-limit system as falling into that 'grey area'.

But as I continue to argue, there is a *vast* literature of
tuning theory, stretching back at least two centuries, which
considers 5-limit JI to be unqualified 'just intonation'.
My opinion is that that usage should be respected.

I think what we really need here is another qualifying term
to characterize the prime/odd limits between 5 and, say, 13.

Perhaps it would be best to retain 'extended' and just add
the limit as a further qualifier for limits between 5 and 43
or thereabouts, e.g., 'Partch's tuning system is 11-limit
extended JI'.

AFAIK, use of 'extended just intonation' originated with Ben
Johnston, who gradually increased the prime-limits in his
compositions from 3, to 5, to 7, to 11, to 13 (where he stayed
for some time), to 19, to 31.

As I've posted here before, Ezra Sims considers his music to
imply ratios up to 37-limit, but he doesn't argue that he should
be called a 'JI composer'. He uses 72-tET notation and considers
his pieces when so tuned (on a computer) to be aurally nearly
indistinguishable from their 37-limit JI realization (on a
computer). And he's more than happy with the results he gets
from live performers, which no doubt deviate considerably from
both of those.

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
>
>
> As I've posted here before, Ezra Sims considers his music to
> imply ratios up to 37-limit, but he doesn't argue that he should
> be called a 'JI composer'. He uses 72-tET notation and considers
> his pieces when so tuned (on a computer) to be aurally nearly
> indistinguishable from their 37-limit JI realization (on a
> computer). And he's more than happy with the results he gets
> from live performers, which no doubt deviate considerably from
> both of those.
>

Monz,

Well! So Ezra Sims considers 37 Prime to be Just Intonation?

I'm wondering if we might have Dave Keenan contact him about this
error?

: )

Jacky Ligon

Carl wrote,

>"A group of pitches is said to be justly intoned when beating cannot
>be heard."

That certainly seems to rule out your

>1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1

since a dyad of 9/8 and 5/3 will beat wildly.

--- In tuning@egroups.com, ligonj@n... wrote:

> http://www.egroups.com/message/tuning/16208
>
> --- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
> >
> >
> > As I've posted here before, Ezra Sims considers his music to
> > imply ratios up to 37-limit, but he doesn't argue that he should
> > be called a 'JI composer'. He uses 72-tET notation and considers
> > his pieces when so tuned (on a computer) to be aurally nearly
> > indistinguishable from their 37-limit JI realization (on a
> > computer). And he's more than happy with the results he gets
> > from live performers, which no doubt deviate considerably from
> > both of those.
> >
>
> Monz,
>
> Well! So Ezra Sims considers 37 Prime to be Just Intonation?
>
> I'm wondering if we might have Dave Keenan contact him about this
> error?
>
> : )

Jacky,

Sims is careful to emphasize that his original reason for
using non-12-tET tuning was that he needed smaller *melodic*
scale-steps than those available in 12-tET. Upon further
study, he realized that the scales that he derived intuitively
happened to have pitches which closely implied harmonic
ratios not well represented in 12-tET.

He doesn't like to stress that he uses 'just intonation' or
any other particular set tuning. He tends to compose pretty
much intuitively (that is, not by setting out a harmonic
or tuning system in advance, other than his 'basic scale'),
and says that the harmonies he hears in his head as he composes
tend to correspond well to the mathematics of summation
and difference tones.

I think the main reason he stops at 37-limit is because it
fit in with the scale he derived melodically - i.e., it's
close to the 'quarter-tone' between 9/8 (the 36th harmonic)
and 19/16 (the 38th).

Sims's major theoretical work is a paper appearing in the
issue of _Computer Music Journal_ devoted to microtonality
(1987). There is another good article in _Perspectives of
New Music_. Check the List Archives: I'm certain that I've
given full citations for his papers before.

I've also given a pretty good description of Sims's tuning
practice in my own book, which you have. It's pretty close
to the end of the book.

(But beware: I've found that my copyist made a mistake, so that
the musical example illustrating either his scale or the one he
analyzed for Louis Armstrong has a missing note-head, which
shifts all the numerical descriptions out of place and makes
them wrong. I'm pretty sure that this error was with the
Armstrong scale and that the Sims scale is OK... I don't have
a copy of my book with me to check. Everyone who has a copy
of the 2nd and 3rd editions should note this error.)

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
> Jacky,
>
>
> Sims is careful to emphasize that his original reason for
> using non-12-tET tuning was that he needed smaller *melodic*
> scale-steps than those available in 12-tET. Upon further
> study, he realized that the scales that he derived intuitively
> happened to have pitches which closely implied harmonic
> ratios not well represented in 12-tET.
>
> He doesn't like to stress that he uses 'just intonation' or
> any other particular set tuning. He tends to compose pretty
> much intuitively (that is, not by setting out a harmonic
> or tuning system in advance, other than his 'basic scale'),
> and says that the harmonies he hears in his head as he composes
> tend to correspond well to the mathematics of summation
> and difference tones.
>
> I think the main reason he stops at 37-limit is because it
> fit in with the scale he derived melodically - i.e., it's
> close to the 'quarter-tone' between 9/8 (the 36th harmonic)
> and 19/16 (the 38th).

Monz,

This is exceedingly fascinating to me that Ezra and I go to 37 prime
for melodic reasons. Very interesting! Tis a sweet spot for many I
see.

>
>
> Sims's major theoretical work is a paper appearing in the
> issue of _Computer Music Journal_ devoted to microtonality
> (1987). There is another good article in _Perspectives of
> New Music_. Check the List Archives: I'm certain that I've
> given full citations for his papers before.

Would like to read this - thanks!

>
> I've also given a pretty good description of Sims's tuning
> practice in my own book, which you have. It's pretty close
> to the end of the book.

I'll be looking tonight.

Jacky

[Monz wrote:]
>Well... I see your point, which is precisely why I mentioned
>Partch's 11-limit system as falling into that 'grey area'.
>
>But as I continue to argue, there is a *vast* literature of
>tuning theory, stretching back at least two centuries, which
>considers 5-limit JI to be unqualified 'just intonation'.
>My opinion is that that usage should be respected.
>
>
>I think what we really need here is another qualifying term
>to characterize the prime/odd limits between 5 and, say, 13.

OK, how about:

. low-limit JI: 1 to 5
. intermediate-limit JI: 7 - 15(?)
. high-limit or extended JI: above 15(?)

I would not make 7-15 a part of "extended JI" because it is audible in
harmonic voices, where larger intervals are not.

JdL

David Keenan wrote:

> I maintain that barbershop singing _epitomises_ JI and can accurately be
> described as 19-limit JI.

I find this very puzzling. As someone who has sung a bit of barbershop, I would say that it
*aims* at 7-limit JI on a chord-to-chord basis. I suppose you're including 19 to represent
tempered minor thirds. But then you've got to either explicitly omit 11 and 13 or else
demonstrate that barbershop makes use of them. I'm not aware of any 11:4s or 13:8s in
barbershop music. If I'm right that your 19-limit analysis is made to account for chords such
as those approximating 16:19:24, then I would say it's not necessary to use numbers that high
for them. I think they should be analyzed as 1/(6:5:4) chords where the minor third is
tempered to reduce melodic "pain." That still makes for JI in my book, since the chief goal is
to make things beatless, even though that goal is balanced with other considerations. I don't
think I would say that it epitomizes JI, though. The epitome would be music in which the
exclusive goal was to make chords beatless, no compromises allowed.

But in principal, I agree with the rest of that post, as far as it goes (the edited version of
the one originally sent off-list to Margo). Margo said it very well not long ago: it's a
matter of first principles. Some of us (Margo and I, it seems, perhaps some others) believe
that the math came first, and existential matter followed it. The validity of that premise is
a matter for another list, but the premise itself affects how we want to define certain terms
on this list.

I especially agree with you, Dave, that a single, universally accepted definition of JI should
be achieved. I'm willing to let "JI" be defined existentially (your definition), and use the
term "rational intonation," for now, for "tuning by integer ratios." RI. It's an important
enough distinction to warrant separate terms.

--
David J. Finnamore
Nashville, TN, USA
http://personal.bna.bellsouth.net/bna/d/f/dfin/index.html
--

David Finnamore wrote,

>If I'm right that your 19-limit analysis is made to account for chords such
>as those approximating 16:19:24, then I would say it's not necessary to use
numbers that high
>for them. I think they should be analyzed as 1/(6:5:4) chords where the
minor third is
>tempered to reduce melodic "pain."

David, we've seen that in a capella singing, and probably in some barbershop
perfomances, 16:19:24 is often a "target" of the minor triad _apart_ from
any melodic considerations (in fact, melodic considerations would usually
weigh _against_ such a tuning. The reason seems to be a virtual fundamental
four octaves below the root.

--- In tuning@egroups.com, ligonj@n... wrote:

> http://www.egroups.com/message/tuning/16213

> --- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
>
> > Jacky,
> >
> >
> > I think the main reason he [Sims] stops at 37-limit is because
> > it fit in with the scale he derived melodically - i.e., it's
> > close to the 'quarter-tone' between 9/8 (the 36th harmonic)
> > and 19/16 (the 38th).
>
> Monz,
>
> This is exceedingly fascinating to me that Ezra and I go to 37
> prime for melodic reasons. Very interesting! Tis a sweet spot
> for many I see.

Hmmm... then you'll be particularly interested to know this
too, Jacky:

In one of my most popular little pieces (at least with List
subscribers), _24-Equal Tune_

http://www.ixpres.com/interval/monzo/worklist/24eqtune.mid

- which is tuned strictly in quarter-tones (= 24-tET) - in the
violin part, I use 2^(5/24) [= exactly 250 cents] melodically
in the 4th measure of the tune in a way that I've always thought
of as representing 37/32 [= just over 251&1/3 cents].

I recall that I wrote this piece right at the time that I was
doing my intensive research on Sims, and that one specific
reason why I chose 24-tET was because I wanted to try to use
37/32 (or a close equivalent) in a piece, after learning that
Sims used it as the prime-limit on his (implied) scale.

(Another reason was because 24-tET has been so maligned by
so many tuning theorists, and I've always thought it more
worthy than many theorists whose work I've read. So I kind
of wrote this piece to prove to myself that one could create
good 24-tET 'pop' music.)

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

--- In tuning@egroups.com, "John A. deLaubenfels" <jdl@a...> wrote:

> http://www.egroups.com/message/tuning/16214

> [Monz wrote:]
> > Well... I see your point, which is precisely why I mentioned
> > Partch's 11-limit system as falling into that 'grey area'.
> >
> > But as I continue to argue, there is a *vast* literature of
> > tuning theory, stretching back at least two centuries, which
> > considers 5-limit JI to be unqualified 'just intonation'.
> > My opinion is that that usage should be respected.
> >
> >
> > I think what we really need here is another qualifying term
> > to characterize the prime/odd limits between 5 and, say, 13.
>
> OK, how about:
>
> . low-limit JI: 1 to 5
> . intermediate-limit JI: 7 - 15(?)
> . high-limit or extended JI: above 15(?)
>
> I would not make 7-15 a part of "extended JI" because it is
> audible in harmonic voices, where larger intervals are not.

As I said, John, I tend to agree with you. I like the idea
of using simply 'intermediate JI' for 7- to 15-odd-limit,
or 7- to 13-prime-limit, and 'extended JI' for higher limits
... and of course, 'rational tuning' for cases of rational
tuning where the concept of JI is not part of the compositional
intention.

But I think 'low-limit JI' is superfluous, because:

1-limit means nothing for musical purposes: it's all unisons.

2-limit likewise only gives 'octaves'.

3-limit already has another name: 'Pythagorean'.

5-limit already has a long history of being called plain old
'just intonation'.

And don't forget that we coined a term here a couple of years
ago that should be part of this discussion: 'WAFSO-just', which
euphemistically stands for 'within a fly's excrement of just'.

-monz
http://www.ixpres.com/interval/monzo/homepage.html
'All roads lead to n^0'

--- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
>
>
> Hmmm... then you'll be particularly interested to know this
> too, Jacky:
>
>
> In one of my most popular little pieces (at least with List
> subscribers), _24-Equal Tune_
>
> http://www.ixpres.com/interval/monzo/worklist/24eqtune.mid
>
> - which is tuned strictly in quarter-tones (= 24-tET) - in the
> violin part, I use 2^(5/24) [= exactly 250 cents] melodically
> in the 4th measure of the tune in a way that I've always thought
> of as representing 37/32 [= just over 251&1/3 cents].

Sounds cool Joe! I'm playing it on my work PC sound card - nifty bass
line!

>
> I recall that I wrote this piece right at the time that I was
> doing my intensive research on Sims, and that one specific
> reason why I chose 24-tET was because I wanted to try to use
> 37/32 (or a close equivalent) in a piece, after learning that
> Sims used it as the prime-limit on his (implied) scale.

Isn't it great where "RI" can lead?

>
> (Another reason was because 24-tET has been so maligned by
> so many tuning theorists, and I've always thought it more
> worthy than many theorists whose work I've read. So I kind
> of wrote this piece to prove to myself that one could create
> good 24-tET 'pop' music.)
>

I thought it was very interesting a while back when Joseph Pehrson
mentioned how well 24 tET has been assimilated into the classically
oriented performer's vocabulary. It's all good. I think lots of folks
must try 24 tET for their first microtonal experiences - for me it
was 36 tET, then 19 and 31.

Jacky Ligon

>
>
> -monz
> http://www.ixpres.com/interval/monzo/homepage.html
> 'All roads lead to n^0'

[Monz:]
>But I think 'low-limit JI' is superfluous, because:
>
>1-limit means nothing for musical purposes: it's all unisons.
>
>2-limit likewise only gives 'octaves'.
>
>3-limit already has another name: 'Pythagorean'.
>
>5-limit already has a long history of being called plain old
>'just intonation'.

But... if you have no qualifier at all on "just intonation" meaning
"historical (5-limit) just intonation", then there is a kind of semantic
implication or expectation which is not being met: that all the
qualified JI's fall under (i.e. are subsets of) the umbrella term "just
intonation" (which is not the case as you define it!).

If you are determined to omit any qualifier on the 5-limit meaning,
it would seem reasonable at least for the definition to resolve any
confusion which might otherwise result.

JdL

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

http://www.egroups.com/message/tuning/16219

>>
> And don't forget that we coined a term here a couple of years
> ago that should be part of this discussion: 'WAFSO-just', which
> euphemistically stands for 'within a fly's excrement of just'.
>

Another possibility could be, unfortunately, "Rational Alternate
Tuning System" or, well... RATS

______________ ___ __ _
Joseph Pehrson

--- In tuning@egroups.com, "Joseph Pehrson" <pehrson@p...> wrote:
> --- In tuning@egroups.com, " Monz" <MONZ@J...> wrote:
>
> http://www.egroups.com/message/tuning/16219
>
> >>
> > And don't forget that we coined a term here a couple of years
> > ago that should be part of this discussion: 'WAFSO-just', which
> > euphemistically stands for 'within a fly's excrement of just'.
> >
>
> Another possibility could be, unfortunately, "Rational Alternate
> Tuning System" or, well... RATS

Or alternately: Prime Extended Systems of Tuning

PEST

: )

Jacky Ligon

>
> ______________ ___ __ _
> Joseph Pehrson

Jacky Ligon wrote:

> P.S. Kyle Gann, is it not correct that you define JI as tuning by
> integer ratios?

Here's Kyle Gann's informal definition. Notice that he nowhere says
that all rational tunings are JI. But he does say all JI tunings are
rational.

In his description of 13 and higher limit JI it seems likely that all
the examples use otonalities with a number of notes proportional to
the size of the numbers involved.

http://www.newmusicbox.org/third-person/sep00/just.html

Regards,
-- Dave Keenan

On Mon, 04 Dec 2000 03:17:27 -0000, ligonj@northstate.net wrote:

>Playing on a Santoor. After about 10 seconds I might return to the
>simple mode again for a reiteration of it's "mood" for another 15
>seconds, when I might introduce completely new timbres and a new
>tuning, such as:
>
> 0: 1/1 0.000
> 1: 206720/180879 231.184
> 2: 221792/180879 353.019
> 3: 1248926/904395 558.796
> 4: 1378112/904395 729.202
> 5: 103360/60293 933.139
> 6: 1754956/904395 1147.692
> 7: 1798022/904395 1189.663

Can anyone really hear the 0.01 cents difference between 206720/180879 and
8/7? Try this and see if you can hear any difference at all:

1/1 8/7 103/84 29/21 32/21 12/7 163/84 167/84

>Now the questions:
>
>1. Is my composition in Just Intonation?
>
>2. Or is only a part of it in JI?

Just as a composition in C major might wander off into other major and
minor keys, I'd argue that a composition in JI doesn't need to contain
exclusively JI intervals.

>3. At what point (prime - odd - whatever) do I cross the barrier of
>complexity and disqualify my composition from the label of "JI"?

I think the near-29/21 is probably the first "questionable" interval in
this scale. It would have to be a very unusual situation to hear anything
like 103/84 as just. It's so close to 11/9, without being close enough to
sound in tune.

>4. How do I reconcile the fact that my piece has a range of simple to
>complex integer ratios, sometimes "justly intoning" beatless
>intervals, and alternately injecting tuning shifts to more complex
>structures, which might not fit the text book classification of JI?
>
>5. What the heck do you call music that morphs it's tunings in this
>manner, if not JI?

You can call it JI if you like, but I think that would be misleading. If
you need something more specific than microtonal, perhaps you might
consider qualifying JI with an adjective ... mostly JI, partly JI, somewhat
JI, nearly JI? Or just call it something like "flexible tuning".

--
see my music page ---> ---<http://www.io.com/~hmiller/music/music.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

Jacky Ligon wrote,

>>when I might introduce completely new timbres and a new
>>tuning, such as:

>> 0: 1/1 0.000
>> 1: 206720/180879 231.184
>> 2: 221792/180879 353.019
>> 3: 1248926/904395 558.796
>> 4: 1378112/904395 729.202
>> 5: 103360/60293 933.139
>> 6: 1754956/904395 1147.692
>> 7: 1798022/904395 1189.663

Herman Miller wrote,

>Can anyone really hear the 0.01 cents difference between 206720/180879 and
>8/7? Try this and see if you can hear any difference at all:>

>1/1 8/7 103/84 29/21 32/21 12/7 163/84 167/84

Notice that the denominators are all power-of-two divisors of 84. It seems
obvious that Jacky is basing this "timbre-tuning" on an FFT of insufficient
resolution -- in this case, a resolution only 84 times finer than the period
of his "1/1", but then using ratios with far too many digits to represent
this underlying pattern. Jacky?

On Mon, 04 Dec 2000 15:52:02 -0000, " Monz" <MONZ@JUNO.COM> wrote:

>To my mind, a composer like Partch, whose stipulated tuning
>system was 11-limit (prime or odd), and who referred to it as
>'just', represents the grey area between these attempts at
>precision. But please note that Partch was also careful to
>employ the term 'Monophony' to describe his specific tuning
>system.

Hmm.... would you consider an 11-limit pentatonic scale "just"?
Specifically:

1/1 11/10 6/5 3/2 8/5

This particular scale happens to be a subset of the harmonic series. So it
looks like it ought to qualify as a JI scale, and I think it sounds like
one. But if you just had the 11/10 without all the other harmonics to
reinforce it, say something like this:

1/1 11/10 5/4 7/5 5/3

Now I just came up with this one on the spur of the moment and haven't
listened to it, so I don't know if the 11/10 makes any kind of sense in
this scale. But it seems to me that this would be a marginal JI scale at
best, or a hybrid scale of mostly JI with one dissonant note.

--
see my music page ---> ---<http://www.io.com/~hmiller/music/music.html>--
hmiller (Herman Miller) "If all Printers were determin'd not to print any
@io.com email password: thing till they were sure it would offend no body,
\ "Subject: teamouse" / there would be very little printed." -Ben Franklin

>1/1 11/10 5/4 7/5 5/3

>Now I just came up with this one on the spur of the moment and haven't
>listened to it, so I don't know if the 11/10 makes any kind of sense in
>this scale. But it seems to me that this would be a marginal JI scale at
>best, or a hybrid scale of mostly JI with one dissonant note.

11/10 could be connected with 7/5 using the marginally tunable interval
7:11.

--- In tuning@egroups.com, "Dave Keenan" <D.KEENAN@U...> wrote:

> http://www.egroups.com/message/tuning/16239
>
> Jacky Ligon wrote:
>
> > P.S. Kyle Gann, is it not correct that you define JI as tuning by
> > integer ratios?
>
> Here's Kyle Gann's informal definition. Notice that he nowhere
> says that all rational tunings are JI. But he does say all JI
> tunings are rational.
>
> In his description of 13 and higher limit JI it seems likely
> that all the examples use otonalities with a number of notes
> proportional to the size of the numbers involved.
>
> http://www.newmusicbox.org/third-person/sep00/just.html

Kyle Gann studied with Ben Johnston; I'd guess that that has
something to do with Kyle's definition of JI.

-monz

>>"A group of pitches is said to be justly intoned when beating cannot
>>be heard."
>
>That certainly seems to rule out your
>
>>1/1 9/8 5/4 4/3 3/2 5/3 15/8 2/1
>
>since a dyad of 9/8 and 5/3 will beat wildly.

That scale isn't mine, nor do I ever suggest calling it a justly intoned
scale. I said that historically, it has been called "just intonation".
Certainly, I never suggested it be treated as a sonority, which is what
the quoted definition applies to, per Dave Keenan's request.

BTW, you must have missed my earlier correction on the Genesis/a*b issue.

-Carl

--- In tuning@egroups.com, Herman Miller <hmiller@I...> wrote:
>
> Can anyone really hear the 0.01 cents difference between
206720/180879 and
> 8/7? Try this and see if you can hear any difference at all:
>
> 1/1 8/7 103/84 29/21 32/21 12/7 163/84 167/84

Herman,

Hello!

Yes, the ratios I listed are really an unimportant byproduct of
converting pitch to cents. Only interesting to me in showing the
subtle variations away from lower ratios. It has been suggested that
I need not post those when discussing the spectrum scales - perhaps a
good idea.

>
>
> Just as a composition in C major might wander off into other major
and
> minor keys, I'd argue that a composition in JI doesn't need to
contain
> exclusively JI intervals.

A good point, since one could achieve justly intoned intervals with
higher primes too if desired.

>
> You can call it JI if you like, but I think that would be
misleading. If
> you need something more specific than microtonal, perhaps you might
> consider qualifying JI with an adjective ... mostly JI, partly JI,
somewhat
> JI, nearly JI? Or just call it something like "flexible tuning".
>

I have decided to adopt the suggestions of Margo and Monz, by using
the more appropriate terms "RI" or Rational Tuning.

Kind Thanks,

Jacky Ligon

--- In tuning@egroups.com, "Paul H. Erlich" <PERLICH@A...> wrote:
>
> Notice that the denominators are all power-of-two divisors of 84.
It seems
> obvious that Jacky is basing this "timbre-tuning" on an FFT of
insufficient
> resolution -- in this case, a resolution only 84 times finer than
the period
> of his "1/1", but then using ratios with far too many digits to
represent
> this underlying pattern. Jacky?

This particular scale is somethng I've been experimenting with, that
is a bit of a variation on the Sethares theme; that of deriving modes
and or chromatic scales from the partials of metal instrument timbres.

My approach to this has been to sort the list of FFT frequencies by
order of amplitude and derive a scale capable of playing in a more
conventional diatonic melodic approach. Something that seems to sound
correct in the context. One could argue (as Paul did point out to me)
that one must not force an octave boundary onto such a scale if
indeed the 2/1 isn't found in the timbre - and this I do generally
agree with.

What seems to make this work on some level to me, could be explained
in the same manner that (many times) harmonic timbres sound correct
with lower number ratios. Something seems to carry over here, where
at least aurally, if I derive a simple mode or scale subset from the
loudest partials, it seems to sound correct with the timbre being
used.

If the loudest partial is for instance on C (65.4064 hz or a cluster
of pitches thereabout), and I form a scale from the other partials of
highest amplitude; when I transpose this on another instrument and to
another octave - it still works - that "C @ 65.4064 hz" provides this
1/1 grounding that carries over (to my ears) from the way we treat
and hear harmonic timbres, In other words I can transpose the tuning
over octaves on another timbre and still sound correct with the
instrument that the scale was derived from. This is trusting what you
hear and not the numbers!

Conversely, I have also just outright mapped my spectrum tunings
across the keyboard without any sorting or reduction - and this is a
totally Xenharmonic explosion of unheard intervals!

As far as your resolution question, I believe it was set to 65,000,
but I won't be able to check until I get back to my studio. But
really, my compostional scenario, was just that - just to find out
what to call this kind of approach.

Thanks!

Jacky Ligon

Herman wrote,

>> Can anyone really hear the 0.01 cents difference between
>206720/180879 and
>> 8/7? Try this and see if you can hear any difference at all:
>>
>> 1/1 8/7 103/84 29/21 32/21 12/7 163/84 167/84

Jacky wrote,

>Yes, the ratios I listed are really an unimportant byproduct of
>converting pitch to cents. Only interesting to me in showing the
>subtle variations away from lower ratios.

But Jacky, it seems incontrovertible that the ratios Herman posted are the
"true" ratios that would have come from your FFT, and the more complex
ratios you posted are simply a result of rounding error. How else could you
explain the denominators all being power-of-two divisors of 84?
Additionally, your resolution would have been no better than 20 cents at the
bottom of the scale, and 10 cents at the top of the scale, so posting the
last digit of the cents value _before_ the decimal point, let alone all the
digits _after_ the decimal point, was meaningless.

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

> digits _after_ the decimal point, was meaningless.

I'll look into this as soon as I can. Disregard the info. It was just
used as an example anyhow.

I'll endeavor to clean up mmy decmal points for posts.

JL

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

Herman and Paul,

Hello!

It seems as though I posted from my old worksheet from before I
learned how to set the FFT size to 65563 (darn - now deleted!).

Below is a correction for my Burmese Nipple Gong mode:

0.000
230.528
352.680
558.929
685.090
943.001
1147.573
1190.938

Thanks for your watchful eyes,

Jacky Ligon

P.S. Quite a different 5th isn't it?

Jacky Ligon wrote,

>Below is a correction for my Burmese Nipple Gong mode:

>0.000
>230.528
>352.680
>558.929
>685.090
>943.001
>1147.573
>1190.938

>Thanks for your watchful eyes,

>Jacky Ligon

>P.S. Quite a different 5th isn't it?

Yes, I'm surprised it moved almost 45 cents, when the resolution of your
original FFT should have been about 15 cents in this part of the scale.
Well, whatever.

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

Paul,

When I discovered the mistake, I found that the FFT size for the data
was set to 4096 rather than 65536, which would've likely caused this
much deviation. The "729.202" was not found(!); proving how important
these resolution settings are.

Thanks again,

Jacky