Yahoo Tuning Groups Ultimate Backup tuning question about just intonation scales

Aline,

Well, you asked an easy one and a couple of harder ones!

A Just Intonation scale is one that consists entirely of
notes whose frequencies are rational multiples of each other.

For example,
A 220.00 Hz 1/1
B 247.50 Hz 9/8
C# 275.00 Hz 5/4
D 293.33' Hz 4/3
E 330.00 Hz 3/2
F# 371.25 Hz 27/16
G# 412.50 Hz 15/8
A' 440.00 Hz 2/1

If you divide any of these frequencies by that of the lowest
note, A, you get a simple fraction, the ratio of one whole
number to another. That makes this a rational scale, and its
notes are said to be in Just (ie exact) Intonation.

That's all there is to being JI.

And that's the easy question answered .... !

Here's the easiest part of the other answers: this scale also
"exhibits tetrachordal similarity". That means it consists of
two tetrachords - sequences of four notes - joined together,
and that the upper tetrachord and the lower one both have
the same interval structure (ratios between notes) internally.

In my example scale of A major, the lower tetrachord is
A B C# D, with interval ratios 1/1 9/8 5/4 4/3. The upper is
E F# G# A', with interval ratios 3/2 27/16 15/8 2/1. If you
divide the ratios for the upper tetrachord by that for its
lowest note E, you get: 1/1 9/8 5/4 4/3 --- exactly the same as
for the lower tetrachord.

Regards,
Yahya

-----Original Message-----

Dear all,

I was wondering what makes a sequence of notes a (just intonation)
scale. Maybe somebody can tell me about properties of just intonation
scales, like conditions they have to satisfy?
In the Encyclopaedia of Tuning I did find some properties of scales.
Under 'scale' is written: "Scales often, but not always, exhibit
tetrachordal similarity, and other properties such as MOS, propriety,
distributional evenness, etc."
However, all these properties are only valid for equal tempered scales
(or am I wrong?).
Are there similar properties for just intonation scales? I hope
somebody can help me. Thanks in advance.

Best regards,
Aline Honingh

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🔗Graham Breed <gbreed@gmail.com>

> I was wondering what makes a sequence of notes a (just intonation)
> scale. Maybe somebody can tell me about properties of just intonation
> scales, like conditions they have to satisfy?

First of, I don't see why they have to be ordered. That is a scale is
a set of notes, not a sequence, and therefore not uniquely tied to
melody. Surprisingly enough, there seems to be disagreement on this.

I'd have thought a scale was in just intonation if all of its
constituent intervals were in just intonation. That leaves the
definition of "just intonation interval" comfortably moot.

> In the Encyclopaedia of Tuning I did find some properties of scales.
> Under 'scale' is written: "Scales often, but not always, exhibit
> tetrachordal similarity, and other properties such as MOS, propriety,
> distributional evenness, etc."
> However, all these properties are only valid for equal tempered scales
> (or am I wrong?).

Tetrachordal similarity is fine for JI, you've had that from Yahya.

MOS is only true for two dimensional scales. That is, scales built up
from two distinct intervals. Just intonation scales usually contain
more than two dimensions. The simplest example of two-dimensional JI
is Pythagorean intonation, but this is rarely described as "JI". If
it isn't JI then my definition above is incomplete.

Rothenberg propriety works for scales of any dimension.

Distributional evennes is almost identical to MOS.

> Are there similar properties for just intonation scales? I hope
> somebody can help me. Thanks in advance.

Just intonation scales can be made consistent with
MOS/distributionally even scales using Fokker periodicity blocks with
two chromatic unison vectors. A less rigorous method would be to look
for step sizes in the JI scale that are almost the same, and consider
them to be equivalent. Continue until you have two step sizes and
then see if the result is MOS/DE. This can be a useful thing to do if
you want to fit a just intonation scale onto a two dimensional
keyboard.

Graham

🔗Jon Szanto <jszanto@cox.net>

🔗Robert Walker <robertwalker@ntlworld.com>

Hi Graham and Jon,

Well a geometric scale, if you think of it as a shape, is unordered. E.g. the ordering by size of a hexany has
no particular significance. You also get scales
that ascend one way and descend the other way.

So, just to say that I'd also go along with Jon,
and say that a scale is an unordered set of pitches
and that the ordering is part of the melodic structure
and that the pitches are ordered for convenience
as a way to project the scale into 1D.
So I'm also a member of that minority school :-).

Robert

> > First of, I don't see why they have to be ordered. That is a scale is
> > a set of notes, not a sequence, and therefore not uniquely tied to
> > melody. Surprisingly enough, there seems to be disagreement on this.

> About the only place you'll find disagreement is in a place like this
> forum. To posit that a scale is some _collection_ of pitches, and is
> not ordered, has to be taken as very much a minority opinion.

🔗Jon Szanto <jszanto@cox.net>

🔗Robert Walker <robertwalker@ntlworld.com>

Sorry that could be unclear.

When I was talking
about trying to arrange the elements of a doubly
infinite scale in increasing order

> In this case it doesn't help at all
> to try ordering the notes in increasing
> order, that only makes the problem
> worse of course they already are ordered
in that doubly infinite one row at a time
way,

> In this case it doesn't help at all
> to try ordering the notes in increasing
> order **of size**, that only makes the problem
> worse because then you could easily end up with
not just an ordering of type omega squared
(infinite rows one at a time one after another)
but even a **dense** ordering ...
(so that there is no single next pitch after any
given pitch at all in the scale).

as before.

Robert

🔗Robert Walker <robertwalker@ntlworld.com>

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Johnny,

You wrote:
-----Original Message-----
If a scale is a mere ordering of notes from low to high from which we
extract
tones to compose, then yes, there can be such a thing as a just intonation
scale.

But really, scale seems more of a metaphor when applied to just intonation.
First off, it is anathema to melody, as Kraig pointed out. For in just
intonation, when all notes are perfectly tapped from the overtone series, a
stupendous blend of harmony ensues. This is not the stuff of a melodic
scale.
...
--------------------------------
I don't think Kraig said, or even implied, that a JI scale is "anathema to
melody".
He may think so, so I'll let him speak for himself. But it's fairly clear
that you
think it is.

Respectfully, I disagree. I can't think of anything that I regard as more
quintessentially melodic than the movement from one note to another a
pure JI interval, a simple ratio, away.

I guess it comes down to a matter of musical style and taste. In a contest
of
taste, there are no winners! :-)

Regards,
Yahya

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🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Gene,

You wrote:

> > --- ... "Robert Walker" ... wrote:
> > Particularly if the scale is
> > an ordering of all possible
> > ratios as some infinite scales are
> > - then it is easy to see that once
> > they are put in ascending order then
> > the result is dense.
>
> I tried to fix that once by defining "scale" so that the notes were
> logarithmically discrete, but people were not happy. I suspect most
> people would prefer that it not be given a strict mathematical
> definition, which of course would also require stating whether it was
> a set, and if so, a set of what.

Speaking only for myself, I'd _love_ a strict mathematical definition,
_provided_ it is capable of encapsulating the essential musical features
as well. In particular, I'd like any definition of a scale to recognise
that it has order as well as content - that a scale is a set of fixed
sequences of pitch classes.

I started a small project, over a month ago now, to define and classify
scales mathematically. But I put it aside as possibly premature, while
soaking up what I could from the members of this and other lists. No
point reinventing the wheel - unless you can do it better! At this point
I'm sure I could only make a square wheel, and have much still to learn.

So, if you DO have a mathematical definition of a scale, I would love to
see it.

Regards,
Yahya

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🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Graham,

Aline wrote:
> I was wondering what makes a sequence of notes a (just intonation)
> scale. Maybe somebody can tell me about properties of just intonation
> scales, like conditions they have to satisfy?

You replied:
> First of, I don't see why they have to be ordered. That is a scale is
> a set of notes, not a sequence, and therefore not uniquely tied to
> melody. Surprisingly enough, there seems to be disagreement on this.
> ...
> Graham

Let me give you an example where the scale has to be a sequence
of notes, not just a set of notes: the melodic minor scale of
common practice. It's usually taught that the scale has two
distinct forms, one ascending and the other descending. For
example, the C minor scale runs thus -

Ascending: C D Eb F G A B C'
Descending: C' Bb Ab G F Eb D C

- and it's in _both_ those forms that countless music students
for centuries have practiced it.

But of course, you knew that! Just thought I'd remind you :-)

.....

From the point of view of _tuning_ the scale, what matters are
the pitches of the notes that comprise it, and their (intervallic)
relationship. The order is a secondary consideration. The set
of notes in any scale, I prefer to call the gamut. In the case of
C minor, we have more than the usual 7 pitches per octave; 9 in
fact. It is an abstraction from the scale. Until historically recent
advances in technology (logarithms, calculators and computers)
simplified calculation, scales have usually arisen from _melodic
practice_, rather than from theoretical harmonic ideas.

Which is not to say that building scales based purely on theoretical
reasons is wrong!; merely that you won't know what kind of music
you can make with them until you hear them in practice. But no
matter what its genesis, any scale is just another musical resource.
Human hearing is the essential medium through which those
resources must pass, and individual taste is the final arbiter of
which resources are useful.

To me, a scale is at least two sequences of notes - one ascending,
another descending - drawn from a gamut, which is a set of notes.

(I say "at least", since in my experience, some Indian singers use
more than one alteration (like that of A Ab in C minor) of a single
scale degree, depending on the particular ornament they're using
at the time. I believe that classically-trained Western violinists
do something similar; whether deliberately or not, I couldn't say.)

Regards,
Yahya

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🔗Afmmjr@aol.com

In a message dated 5/21/2005 8:32:06 AM Eastern Standard Time,
yahya@melbpc.org.au writes:
don't think Kraig said, or even implied, that a JI scale is "anathema to
melody".
He may think so, so I'll let him speak for himself. But it's fairly clear
that you
think it is.
Hi Yahya,

Kraig said: "Scales are concerned with melodic qualites of a series of notes
( as
opposed to harmonic ).:

I read into this that if "scales are concerned with melodic qualities"
basically, then just intonation is a poor representative as a scale in this regard.

Respectfully, I disagree. I can't think of anything that I regard as more
quintessentially melodic than the movement from one note to another a
pure JI interval, a simple ratio, away.
Before we rest the matter as pure opinion, please allow me to elaboraate. In
my experience as a composer using many different systems of and approaches to
tuning, certain intervals demonstrate greater melodic pull than others. For
example, quartertones make for great melodic tension, as does the leading tone
of Romantic style music.

In contradistinction, just intervals -- as beautiful as you may describe
their ascent or descent to another -- by their very nature melt away immediately
into harmony. This is especially so through the 7th limit. The mere movement
of a melodic JI interval to another is overshadowed by their role in a
harmonic connection. As a matter of priority of function, JI is weak in melodic
tension. 13 and 11 limit interval make a huge difference.

I guess it comes down to a matter of musical style and taste. In a contest
of
taste, there are no winners! :-)

Regards,
Yahya
Or it may be based on a wider sampling of what kinds of intervals are
available for musical use. Melodic tunings of a menagerie of variations exist around
the world, some making no JI sense. And yet they work fantastically for
their musical urpose.

Of course there is not contest. We are sharing our understandings of music
in the hopes of approaching truth.

all best, Johnny Reinhard

🔗Afmmjr@aol.com

In a message dated 5/21/2005 8:33:51 AM Eastern Standard Time,
yahya@melbpc.org.au writes:
To me, a scale is at least two sequences of notes - one ascending,
another descending - drawn from a gamut, which is a set of notes.
I have always taught haromnic minor as a 9 note scale, rather than two
different scales based on direction. When analyzing Bach for chord content, it
seemed the more coherent manner. On the other hand, I find it amazing that I name
and file away musical intervals differently when they are ascending than when
they are descending.

For example, an ascending perfect fourth is "Here Comes the Bride," while a
descending perfect fourth is reached directly by the opening inerval of
Schubert's "Unfinished Symphony."

all best, Johnny

🔗Kraig Grady <kraiggrady@anaphoria.com>

This is no tried and true definition of a scale, even a just intonation scale. As i attempted to do was to show a few of the properties , that if present, are heard as a scale.
It seems though that scale differs from structure and is a melodic property. Partch is a good example who used the diamond as a harmonic structure but filled it out to make a 43 tone scale
Erv Wilson who discovered the hexany would likewise not see the hexany as a scale, but would add tones to make who would be some type of melodic integrity
And there are plenty of scales that are not limited to limits, some very ancient like some of Ptolemy, or any of the scales i use, which are just intonation by definition but have nothing to do with limits.
Moments of symmetries only exist in the case of the pythagorean. while there are many constant structures, and these contain many subsets that most would hear as a scale , but do not satisfy being distributed evenly.
There are scales that don't fit into constant structures either such as the enharmonic scales like many tetrachordal scales.
Compositionally i have used what to my ear sounds like a scale by might only span a fifth or a tritone. such compressed scales are common in north africa.
So the more we tried and pin it down, the more it evades us. i am sure that others will come up with other ways of generating scales or might find a property that includes all the above
If we play aa series of ratios and we have no desire to add any pitches to the spectrum, or feel a gap we would like to fill then this is probably a scale.
There is also the case of where we use not all the notes of a scale per se, and here i guess the guide would be that if we added particular notes and we felt it did not change the spectrum or melodic field then we might consider these a part of the scale.
One cannot define scale without a certain amount of subjectivity.
--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗Robert Walker <robertwalker@ntlworld.com>

Hi Gene,

> I tried to fix that once by defining "scale" so that the notes were
logarithmically discrete, but people were not happy. I suspect most
people would prefer that it not be given a strict mathematical
definition, which of course would also require stating whether it was
a set, and if so, a set of what.

Well I'm game to suggest a definition of scale, as a point
for discussion, mainly to show up the difficulties.

How about - a scale is an infinite countable unordered
subset of the positive reals which must have 1 as a member.

Then if someone says that scales are usually finite
- well that is because you usually give them an
interval of repetition. So what you have there
isn't a scale as such, but a scale generator.

A bit like saying
Here are the natural numbers:

1, 2, 3, 4, ...

But you haven't written them all down of course.

Here is a scale

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

It's just that normally we leave out the
... as "understood". There the method we
are using to generate the scale uses
a particular ordering, and of course
has to because that's how it works.

However, the scale we
are using it to generate doesn't have
to be thought of as ordered in that
way - the same scale can be
ordered in whatever way we please.

1/1 4/3 3/2 9/8 5/4 5/3 15/8 2/1 ...
would also generate the same scale
and if anyone wants to use that ordering
or whatever ordering they please, then fine.

That then avoids the problem of getting into such
things as intervals of equivalence and the question
of how to treat non octave scales, or all
the different types of scales there could be
- any kind of scale is allowed by this definition
and you don't need to enumerate them in advance.
It means someone can come up with some completely
new kind of scale construction idea and so long as
it generates countably many positive numbers
then it is included in our mathematical definition
so we won't need to keep updating our definition
to take account of new scale construction ideas.

It does have a bit of a drawback though because
you might want to allow a range of values
for an element in a scale.

More generally, maybe a scale should be
regarded as a set of line segments rather
than a set of points. Normally each line
segment is of zero length so you lose nothing
by treating them as points. But they can
be of non zero length corresponding to a range of values. Someone might want to go
further and say that each line segment
should also have a weighting attached to
it so that e.g. the line segment around
3/2 might have 3/2 itself as the most
3/2 like interval, and then tail off
quite rapidly to either side so that
say 50 cents flat or sharp is very low
probability or zero probability 3/2
(in some particular context).

But the one around 5/4 might have
a more slowly tapered weighting so that
even an 11/9 or 14/11 is within the range of possible 5/4s.

What is your idea about defining a scale
so that its notes are logarithmically
discrete?

Robert

🔗Robert Walker <robertwalker@ntlworld.com>

Hi Yahya

> Could you please give an example of the sort of
doubly infinite scale you had in mind? Your
later comments on "density" suggest you may
have been thinking of a continuum, rather than
a countable infinity.

I was thinking about the Lambdoma
which I've been working with a lot
recently for Barbara Hero's keyboard.
(Also known as the tonality diamond)
.. .. .. .. .. .. .. .. ..
.. 8/8 8/7 8/6 8/5 8/4 8/3 8/2 8/1
.. 7/8 7/7 7/6 7/5 7/4 7/3 7/2 7/1
.. 6/8 6/7 6/6 6/5 6/4 6/3 6/2 6/1
.. 5/8 5/7 5/6 5/5 5/4 5/3 5/2 5/1
.. 4/8 4/7 4/6 4/5 4/4 4/3 4/2 4/1
.. 3/8 3/7 3/6 3/5 3/4 3/3 3/2 3/1
.. 2/8 2/7 2/6 2/5 2/4 2/3 2/2 2/1
.. 1/8 1/7 1/6 1/5 1/4 1/3 1/2 1/1

The thing is that every possible ratio occurs in there somewhere.
So it includes all the rationals. So it is dense. For instance
there is no next ratio after 1/1 because between 1/1 and say
3/2 you have 4/3. Between 1/1 and 4/3 you have 5/4, and so on.
In fact there is no next ratio after any of the pitches in the
scale.

The way that works is through Cantor's diagonal proof that the
number of ratios is the same as the number of whole numbers.
So as you say, the ordinals omega and omega squared are the same in cardinality
and both are the same as the ordering of the rationals (which
isn't an ordinal any more, because it isn't "well ordered" - an ordinal
is an ordering in which every member has a "next member").

Yes you can do a diagonal too.
1/1 8/7 7/8 4/3 (1/1) 3/4 ... leaving out all repetitions
to enumerate the rationals. One could take that as ones
"scale" at least, as a gamut if one needs one that can
be ordered one dimensionally (or as mathematicians say,
"linearly ordered").

Robert

🔗Robert Walker <robertwalker@ntlworld.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@c...> wrote:

/tuning/topicId_58625.html#58630

> --- In tuning@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> > First of, I don't see why they have to be ordered. That is a
scale is
> > a set of notes, not a sequence, and therefore not uniquely tied to
> > melody. Surprisingly enough, there seems to be disagreement on
this.
>
> About the only place you'll find disagreement is in a place like
this
> forum. To posit that a scale is some _collection_ of pitches, and is
> not ordered, has to be taken as very much a minority opinion.
>
> Cheers,
> Jon

***Traditionally, if it's *unordered* it is, of course, considered
a "set" not a "scale..."

🔗Graham Breed <gbreed@gmail.com>

🔗Joseph Pehrson <jpehrson@rcn.com>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@g...> wrote:
> On 5/22/05, Joseph Pehrson <jpehrson@r...> wrote:
>
> > ***Traditionally, if it's *unordered* it is, of course, considered
> > a "set" not a "scale..."
>
> I like that "of course" for something I've never heard of before :-S
>
> The references I checked after Jon corrected me say that "mode" is the
> word for an unordered set of notes. Does that make any sense to you?
> How about "notes of the scale"?
>
>
> Graham

***Hello Graham!

Personally, I believe "modes" are also thought of as having a pitch
order, like scales (at least the traditional "Church Modes" and such
like...)

I believe, in the 20th Century (not the 21st!) the term "set" was
coined just for these purposes: to present discrete pitches together
in a group that was intentionally not ordered, but which could be
subject to various arithmetic transformations...

Sorry about the "of course..." I really wasn't being snide, but I
guess I assumed that people knew about this kind of set theory
stuff... :)

J. Pehrson

🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Hi Johnny,

Thanks for your reply. Comments inserted below.

Regards,
Yahya

[YA] I don't think Kraig said, or even implied, that a JI scale is "anathema
to
melody". He may think so, so I'll let him speak for himself. But it's
fairly clear that you think it is.

[JR] Hi Yahya,

Kraig said: "Scales are concerned with melodic qualites of a series of notes
( as opposed to harmonic ).:

I read into this that if "scales are concerned with melodic qualities"
basically, then just intonation is a poor representative as a scale in this
regard.

[YA] Yes, I thought this was your interpretation, but didn't interpret
Kraig's words that way myself.

[YA, earlier] Respectfully, I disagree. I can't think of anything that
I regard as more quintessentially melodic than the movement from one
note to another a pure JI interval, a simple ratio, away.

[JR] Before we rest the matter as pure opinion, please allow me to
elaborate. In my experience as a composer using many different
systems of and approaches to tuning, certain intervals demonstrate
greater melodic pull than others.

[YA] Yes, I did think this might have been your meaning. Thank you for
clarifying that. I agree with you, based on my own experience as a
vocalist, player, listener and composer. Also, I think that the force
tending to move a particular note to another one (melodic pull) is
greater when the gamut in use provides notes closer to the first.
This is something you can see quite clearly by choosing simpler or
more complex scales, even in 12-EDO or JI.

[JR] For example, quartertones make for great melodic tension, as does
the leading tone of Romantic style music.

In contradistinction, just intervals -- as beautiful as you may describe
their ascent or descent to another -- by their very nature melt away
immediately into harmony. This is especially so through the 7th limit.
The mere movement of a melodic JI interval to another is overshadowed
by their role in a harmonic connection. As a matter of priority of
function, JI is weak in melodic tension. 13 and 11 limit interval make a
huge difference.

[YA] I see what you're saying, but for me - most particularly in
the music I write - the harmonic context is usually the result of
melodic movements, rarely the reverse. The "mere movement"
between JI notes melodically is my primary resource; from it I
build figures (motives, themes) and it's the interplay of these
figures that creates the music. So - in the context of my own
music at least - I can't agree that "just intervals ... by their very
nature melt away immediately into harmony." I suggest that when
they do, it's a result of musical style, rather than of harmonic
nature.

[YA, earlier] I guess it comes down to a matter of musical style
and taste. In a contest of taste, there are no winners! :-)

[JR] Or it may be based on a wider sampling of what kinds of
intervals are available for musical use. Melodic tunings of a
menagerie of variations exist around the world, some making
no JI sense. And yet they work fantastically for their musical
purpose.

[YA] Very true.

[JR] Of course there is not contest. We are sharing our
understandings of music in the hopes of approaching truth.

all best, Johnny Reinhard
[YA] You bet!

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🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Johnny,

I agree that the melodic minor scale uses a 9-note gamut. Its
ascending and descending forms draw different notes from
this gamut. The harmonic minor scale, however, uses only a
7-note gamut. Its ascending and descending forms use the
exact same notes.

Another way of looking at these scales is to say that each
contains 7 scale degrees, some of which - the sixth and
seventh - have alternate forms which we need to use to suit
a particular melodic (or harmonic?) context.

I guess that one reason I think of scales as being ordered
is from long-term interest in classical Indian music, in which
the raga is more than a scale, it is a pattern of melodic
movement, more closely allied to Western ideas of theme
or motif.

I think it perfectly natural that you do remember and "file
away" the rising and falling intervals as different, since the
two directions of motion can have vastly different emotional
effects. For example, the rising fifth that opens "The Star
Wars Theme" is very different in effect from the falling fifth
that begins "The Wild Colonial Boy".

Regards,
Yahya

-----Original Message-----
[YA]
To me, a scale is at least two sequences of notes - one ascending,
another descending - drawn from a gamut, which is a set of notes.

[JR]
I have always taught haromnic minor as a 9 note scale, rather than
two different scales based on direction. When analyzing Bach for
chord content, it seemed the more coherent manner. On the other
hand, I find it amazing that I name and file away musical intervals
differently when they are ascending than when they are descending.

For example, an ascending perfect fourth is "Here Comes the Bride,"
while a descending perfect fourth is reached directly by the opening
interval of Schubert's "Unfinished Symphony."

all best, Johnny

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🔗Yahya Abdal-Aziz <yahya@melbpc.org.au>

Robert,

Thanks for your example (the lambdoma).

Yes, it's certainly dense. And it provides
a neat way of making - for all _practical_
purposes - a musical scale which is JI but
- considered only as a gamut, or set of
notes or intervals - is still indistinguishable
from the set of reals. Any pitch you may
choose to play lies arbitrarily close to some
member of the lambdoma.

It's only one's selection of notes from that
gamut, and their ordering in time, that could
differentiate the lambdoma from a scale
consisting of all notes.

Regards,
Yahya

-----Original Message-----
________________________________________________________________________

Hi Yahya

The way that works is through Cantor's diagonal proof that the
number of ratios is the same as the number of whole numbers.
So as you say, the ordinals omega and omega squared are the same in
cardinality
and both are the same as the ordering of the rationals (which
isn't an ordinal any more, because it isn't "well ordered" - an ordinal
is an ordering in which every member has a "next member").

Robert
________________________________________________________________________

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🔗Kraig Grady <kraiggrady@anaphoria.com>

🔗Gene Ward Smith <gwsmith@svpal.org>

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

> I have already mentioned Partch's example of the Lambdoma where he did
> not concider the diamond a scale, but only the full set of 43 tones
that
> filled in the gaps.
> Once again the hexany by its inventor was and is not concidered a scale

I think any definition has to be general and cannot take into account
personal opinions on particular cases, except to help guide to a
general definition. From that point of view, the real question is
*why* Partch did not consider the 11-limit diamond a scale, but did
consider his 43 tones to constitute a scale. What general principles
are involved, and how can they be formulated explicitly?

As for the hexany, the septimal hexany is epimorphic, constant
structure, and strictly proper. Having six tone-classes, it is in the
magic range for scales whose notes can be recalled and kept in mind.
If it doesn't qualify as a scale, then very little else can or will. I
don't agree that discovering the hexany means you get to decide if it
is a scale or not, but if so, I discovered it also, in the late
sixties. I composed in it in the seventies, when I certainly thought
of it as a scale.

However, if it isn't a scale, how about this baby:

! octone.scl
octone around 49/40-7/4 interval
8
!
15/14
49/40
5/4
10/7
3/2
12/7
7/4
2

It's closely related for theoretical reasons to the hexany, and in
fact contains one. Now, is it a scale or not, and either way, for what
reason?

🔗Kraig Grady <kraiggrady@anaphoria.com>

Hi Gene!
It is true that one could find a hexany that is a scale, but the point i maybe was not clear with is that not all hexanies are automatically scales. same with eikosanies. the 1-3-7-9-11-15 one has to add only two, with the 1-3-5-7-9-11, one has to go out to 31 places.
to make a stable scale melodically.

Through the years on this list , i have found that a feel for what a scale is has been lacking.
Where series of notes and the validity of a gamut of tones is judged solely on it harmonic properties or the number of consonances it has in it. While valid, much is lost if we do not also take into consideration what goes into melodic integrity and consistency.
Where else can one look at and observe this if not in the concept of scale.
like i said, we do not know fully what a scale is, but it is a look at the up and down sequence of a series of pitches.
We know a few things that make good ones, at least most of the time. More will be discovered.
Mankind more often than not will fill in the gaps of larger intervals, not always, and we should not ignore this. Even with pelog , being a good example, the singers us tones in between

the question of the melodic minor is simple if one just view the scales as having tones of variation. a seven tone scale in which two vary or as a nine tone scale with certain rules. I don't see a problem there

exactly like the diamond, tones are added to fill out the gaps to give the "ladder" of tones a certain melodic consistency. Novaro also added tone to make a scale out of the 7 limit diamond. which is a two dimensional view of a series of note up or down.
As far as up and down, one can look at slonimsky's thesaurus of scales and melodic patterns. When one varies from up and down sequence he refers it to a melodic pattern. Message: 10 Date: Mon, 23 May 2005 23:01:35 -0000
From: "Gene Ward Smith" <gwsmith@svpal.org>
Subject: Re: question about just intonation scales

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

>> I have already mentioned Partch's example of the Lambdoma where he did >> not concider the diamond a scale, but only the full set of 43 tones
> >
that >> filled in the gaps.
>> Once again the hexany by its inventor was and is not concidered a scale
> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "ahoningh2000" <ahoningh@s...> wrote:
> Dear all,
>
> I was wondering what makes a sequence of notes a (just intonation)
> scale. Maybe somebody can tell me about properties of just intonation
> scales, like conditions they have to satisfy?
> In the Encyclopaedia of Tuning I did find some properties of scales.
> Under 'scale' is written: "Scales often, but not always, exhibit
> tetrachordal similarity, and other properties such as MOS, propriety,
> distributional evenness, etc."
> However, all these properties are only valid for equal tempered scales
> (or am I wrong?).
> Are there similar properties for just intonation scales? I hope
> somebody can help me. Thanks in advance.
>
> Best regards,
> Aline Honingh

Dear Aline,

I understand you to be asking, "Given a set of pitches in just
intonation, what properties determine whether they can be considered a
scale." Or possibly "a good scale".

Some readers thought you were asking what determines whether a scale
can be considered to be in just intonation. One even thought this was
easy to answer, namely:

"A Just Intonation scale is one that consists entirely of
notes whose frequencies are rational multiples of each other."

I must point out that this definition is utterly useless because the
distinction between rational and irrational is not something that we
can hear or even measure.

To me, a justly intoned interval is one that can be tuned by ear so as
to be nearly beatless. And a just intonation scale is one that can be
tuned by ear without using notes outside the scale. So not every
interval in the scale has to be just, not even every interval that
includes the tonic, but there must be a path from every pitch to every
other pitch, using only just intervals of the scale.

But to answer the other question.

Tetrachordality is just as much a good property for a JI scale to have
as a tempered one. In place of equal or linear temperament scales
being a moment-of-symmetry or being distributionally even, we have the
property for JI scales of being a "periodicity block". Propriety and
strict propriety work for JI scales as well as non-JI. A related but
slightly weaker property that applies to all is that of being a
"constant structure". Another is that of having a cardinality, i.e.
number of notes per octave or other equivalence-interval, that is
between about 5 and 10.

-- Dave Keenan

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Jon Szanto <jszanto@cox.net>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > To me, a justly intoned interval is one that can be tuned by ear so as
> > to be nearly beatless. And a just intonation scale is one that can be
> > tuned by ear without using notes outside the scale.
>
> It seems to me this would make 19 equal a just intonation scale.
> If we start from 220 Hz, then a minor third above it in 19 is
> 264.0226 Hz. The sixth partial of 220 and the fifth partial of
> 264.0226 are just 0.113 Hz apart, leading to beats of 8.85 seconds.

You have a good point there, Gene.

But first, many would say 19 notes is too many to be considered a
scale, so lets talk instead about tunings.

I would have to agree that if 19 notes in a (octave-reduced) chain of
just minor thirds is a JI tuning then so is 19-equal.

So the question is really, why isn't a chain of just minor thirds a JI
tuning?

We could just say that it is a JI tuning, but it isn't a very
interesting one since it only has one kind of just interval (and its
octave-inversions and extensions).

Or we could say that for a tuning to be called JI it needs to be more
strongly connected by just intervals (i.e. intervals tunable by ear)
than a merely linear chain (or cycle), which would also eliminate the
concept of Pythagorean as "3-limit JI".

-- Dave Keenan

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Jon Szanto <jszanto@cox.net>

🔗Kraig Grady <kraiggrady@anaphoria.com>

Once again i agree with Dave Keenan here.
except that i would not call 'constant structure ' a weaker property since it is what you have to have if you have more than 2 interval sizes. It is formed by JI intervals mapping to an MOS archetype as one way to see it. But i am biased, Constant structures are my favorite type of scales. Partches 43 tone scale interpeted in a 41 tone context is a constant structure, as well as the 31 tone mapping of the 1-3-5-7-9-11 eikosany, the 22 tone 1-3-7-9-11-15 eikosany, and some of the euler genus.
These all i would not consider having 'weaker properties'.

Dear Aline,

I understand you to be asking, "Given a set of pitches in just
intonation, what properties determine whether they can be considered a
scale." Or possibly "a good scale".

Some readers thought you were asking what determines whether a scale
can be considered to be in just intonation. One even thought this was
easy to answer, namely:

"A Just Intonation scale is one that consists entirely of
notes whose frequencies are rational multiples of each other."

I must point out that this definition is utterly useless because the
distinction between rational and irrational is not something that we
can hear or even measure.

But to answer the other question.

-- Dave Keenan

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗Ozan Yarman <ozanyarman@superonline.com>

Brother, I like the way you put it. If I may be allowed to rephrase, a gamut is an ascending-descending order of dynamic notes, each sounded one after the other, as compared to a scale which is merely a static set of tones (confined in a period) from low to high. So, the `melodic minor` is a gamut made up of 9 distinct pitches, whose scale has the 6th and 7th degrees alterated by a half-tone upon descent.

And a minor key would have all instances of harmonic, melodic and natural minors within it. I am hereby inclined to say that maqams (and maybe rags and destgahs too) are actually keys per se.

Cordially,
Ozan

From: Yahya Abdal-Aziz
To: tuning@yahoogroups.com
Sent: 23 Mayıs 2005 Pazartesi 7:37
Subject: [tuning] Re: question about just intonation scales

Johnny,

Regards,
Yahya

-----Original Message-----
[YA]
To me, a scale is at least two sequences of notes - one ascending,
another descending - drawn from a gamut, which is a set of notes.

For example, an ascending perfect fourth is "Here Comes the Bride,"
while a descending perfect fourth is reached directly by the opening
interval of Schubert's "Unfinished Symphony."

all best, Johnny

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🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, Kraig Grady <kraiggrady@a...> wrote:
> Once again i agree with Dave Keenan here.
> except that i would not call 'constant structure ' a weaker property
> since it is what you have to have if you have more than 2 interval
> sizes.

Hi Kraig,

I meant that being a constant structure (CS) was a weaker property
than being (Rothenberg) proper, not weaker than being MOS or DE.

Rothenberg propriety can be applied to scales with more than two
interval sizes and there are scales which are CS but are not
Rothenberg proper e.g. Blackjack. If however there are scales which
are proper but not CS then I would be wrong to call CS weaker than
propriety.

I suspect that all your favourite CS scales are also proper.

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@c...> wrote:
> --- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> > It seems to me this would make 19 equal a just intonation scale.
> > If we start from 220 Hz, then a minor third above it in 19 is
> > 264.0226 Hz. The sixth partial of 220 and the fifth partial of
> > 264.0226 are just 0.113 Hz apart, leading to beats of 8.85 seconds.
>
> OK, let me ask: have you ever tuned up an instrument, or even two
> strings, by tuning the 6th and 5th partials of the given notes? IOW,
> don't you think the "tuning by ear" that is being discussed is
> primarily speaking of tuning the fundamentals of the pitches?
>
> Jon

Hi Jon,

I'm not sure what you're getting at here. I was speaking of tuning
using the whole timbre, not just the fundamental. But for my
definition, I don't really mind if, on a stringed instrument, the
tuning has to be done by damping strings at the appropriate nodes to
reduce all partials but the ones whose beating we are interested in.
I've tuned a 5:6 that way on a guitar, even a 6:7, but beyond that I'd
need new strings and something smaller than my finger to damp the node
with. Then maybe I'd hear a loud enough 11th harmonic to tune against,
although I expect it would still decay very rapidly, so it would be
difficult.

-- Dave

🔗Afmmjr@aol.com

In a message dated 5/23/2005 11:58:18 AM Eastern Standard Time,
kraiggrady@anaphoria.com writes:
More cultures use just intonation melodically than harmonically. , India
and the mid east. I can not think of any that use it harmonically.
Actually, mid-east music is rarely considered to be in just intonation
(except maybe Turkish as 3-limit, and some forms of Iranian). Besides, these are
particularly melodic systems.

Harmonic examples of JI could include Sardinian vocal music and Kung! music
from South Africa.

Besides, Kraig -- as you point out -- much of the determination of what is
harmonic and what is not is in the mind's ear.

all best, Johnny Reinhard

p.s. in ancient Greece harmony and melody were reversed!

If anything JI gives a melodic richness that Et do not, while the
latter might be argued more useful for harmony.
Lou Harrison found JI his inspiration for melodic material

🔗Ozan Yarman <ozanyarman@superonline.com>

Dear Johnny,

It is with great regret and consternation that I admit past and current theories on Middle Eastern Music tradition dismiss the just intonation nature of maqams! For indeed, pythagorean cycle of fifths and quarter-tone systems did and still do a great disservice to the innate beauty of Maqam Music. Fortunately, renown practitioners seldom abide by the book when performing it. It is incumbent upon me then, to re-evaluate theory and provide a well-grounded foundation to resolve this mess.

Cordially,
Ozan

----- Original Message -----
From: Afmmjr@aol.com
To: tuning@yahoogroups.com
Sent: 25 Mayıs 2005 Çarşamba 5:21
Subject: Re: [tuning] RE: question about just intonation scales

Actually, mid-east music is rarely considered to be in just intonation (except maybe Turkish as 3-limit, and some forms of Iranian). Besides, these are particularly melodic systems.

Harmonic examples of JI could include Sardinian vocal music and Kung! music from South Africa.

Besides, Kraig -- as you point out -- much of the determination of what is harmonic and what is not is in the mind's ear.

all best, Johnny Reinhard

p.s. in ancient Greece harmony and melody were reversed!

🔗Jon Szanto <jszanto@cox.net>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> My one experience trying to tune a piano was not a happy one. Don't
> you think this is a little rude? My point does not depend on how good
> a piano tuner I am, and thinking you can refute what I say by pointing
> to my personal inadequacies is a rhetorical ploy which has long ago
> become tiresome.

Without reading any further follow-ups (I've got dinner guests and I
snuck away for a moment), I'll just say this: if I completely
misunderstood your proposition on a 19et scale being JI because of
some tiny tuning between partials...

It seemed like you were *actually* basing an example by looking at the
difference between the 5th and 6th partials of two pitches, which you
determined could be tuned with a beating period of something like 8
seconds. And what I really wanted to know is if you expected in the
real world to be able to - by ear - isolate pitches a number of steps
up in the partial series for tuning, and using just that criteria to
determine the justness of a scale.

Because if it *can't* be done, if it is so very much outside of the
ballpark (sorry for the 'Americanism', folks), then your commentary is
simply a clever, albeit unhelpful, play on the intricacies of the
measurements. It is like arguing with a defense attorney, who will
keep bringing up more and more arcane - and irrelevant - points of
law, just because they are clever.

You _know_ 19et isn't a just system, right?

And, seriously Gene, if I somehow misconstrued your example of
partials, you have my apology. Any rudeness would only apply if you
were, indeed, flaunting microscopic calculations over any ability for
a human being to tune it. As in a musical situation.

Jon

🔗Jon Szanto <jszanto@cox.net>

--- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
> I'm not sure what you're getting at here. I was speaking of tuning
> using the whole timbre, not just the fundamental. But for my
> definition, I don't really mind if, on a stringed instrument, the
> tuning has to be done by damping strings at the appropriate nodes to
> reduce all partials but the ones whose beating we are interested in.
> I've tuned a 5:6 that way on a guitar, even a 6:7, but beyond that I'd
> need new strings and something smaller than my finger to damp the node
> with. Then maybe I'd hear a loud enough 11th harmonic to tune against,
> although I expect it would still decay very rapidly, so it would be
> difficult.

Right, Dave. I did infer from Gene's post that he *was* talking about
comparing two notes that happened to be at the 5th and 6th partials,
respectively, of two other notes. And whether or not the ability to do
that was within the realm of reason. I find that if one were to go on
simply - and arbitrarily - mechanical or mathematical upper divisions
of a timbre, one could eventually say _everything_ was just. Which, to
my mind, is playing a game and not being honest about the musical
intent of a JI scale as opposed to an ED scale.

But then a scale doesn't _have_ an intent, does it?

Cheers,
Jon

🔗Gene Ward Smith <gwsmith@svpal.org>

--- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@c...> wrote:

> Right, Dave. I did infer from Gene's post that he *was* talking about
> comparing two notes that happened to be at the 5th and 6th partials,
> respectively, of two other notes.

I was talking about Dave Keenan's proposed defintion of JI scale, and
pointing out that 19 equal is connected by near-just 6/5 and 5/3
ratios. Your response to that almost seemed to amount to an assertion
that one can only tune octaves by ear.

And whether or not the ability to do
> that was within the realm of reason.

Why is tuning a 3:5 ratio by ear not within the realm of reason, and
if it isn't, what is?

🔗Jon Szanto <jszanto@cox.net>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> I was talking about Dave Keenan's proposed defintion of JI scale, and
> pointing out that 19 equal is connected by near-just 6/5 and 5/3
> ratios.

A tenuous connection, but we'll leave it there.

> Your response to that almost seemed to amount to an assertion
> that one can only tune octaves by ear.

You misunderstood. It happens.

> And whether or not the ability to do
> > that was within the realm of reason.
>
> Why is tuning a 3:5 ratio by ear not within the realm of reason, and
> if it isn't, what is?

Then it was *my* understanding, or your opacity. For the third time,
it seemed like your wording implied that you were listening to the
beating between the 5th particial of one pitch against the 6th partial
of another pitch. It seemed a stretch to expect people to tune in that
manner.

No harm intended.

Jon

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Jon Szanto" <jszanto@c...> wrote:
> Right, Dave. I did infer from Gene's post that he *was* talking about
> comparing two notes that happened to be at the 5th and 6th partials,
> respectively, of two other notes. And whether or not the ability to do
> that was within the realm of reason.

Ah. No. Gene's point is that every note of 19-equal can be reached
from every other note by a chain of just minor thirds. Change
"19-equal" to "19 notes of 1/3-comma meantone" if you like, they are
audibly indistinguishable. It produces the same quandry for my
definition of a JI scale. If one admits of Pythagorean as being JI,
then 19-equal is also JI in a similar fairly-uninteresting sort of way.

But I note that the same thing holds for 1/4-comma meantone, since
every note can be reached from every other note by a chain of just
major thirds. But no-one wants to call this JI. "Quasi-JI" is the best
it gets. So I'm inclined to say the same about 1/3-comma and
Pythagorean. Except that Pythagorean might be considered JI because as
well as 2:3's it also contains 4:9 or 8:9 intervals which are directly
tunable by ear.

That means I need to tweak my JI scale definition to say that every
note should be reachable from every other note, _and_ the number of
just intervals should be greater than the number of notes in the scale
(assuming octave-equivalence).

> But then a scale doesn't _have_ an intent, does it?

Some have maintained that the intent of the designer of a scale _can_
determine whether or not it is JI. This seems like bad move to me. It
seems to me that one ought to be able to determine whether a scale is
just or not by tuning it up on an instrument (let the scale designer
specify the instrument) and doing various listening tests on it,
including being allowed to slightly vary the tuning of notes and see
how much it affects the sounds of the just harmonies (let the scale
designer specify which harmonies, or horizontal intervals).

And of course we would have to admit of degrees of justness. Some
scales would be heard as more just than others.

-- Dave Keenan

🔗Kraig Grady <kraiggrady@anaphoria.com>

________________________________________________________________________

Message: 20 Date: Tue, 24 May 2005 22:21:31 EDT
From: Afmmjr@aol.com
Subject: Re: RE: question about just intonation scales

In a message dated 5/23/2005 11:58:18 AM Eastern Standard Time, kraiggrady@anaphoria.com writes:
More cultures use just intonation melodically than harmonically. , India and the mid east. I can not think of any that use it harmonically.
Actually, mid-east music is rarely considered to be in just intonation (except maybe Turkish as 3-limit, and some forms of Iranian). Besides, these are particularly melodic systems.

Harmonic examples of JI could include Sardinian vocal music and Kung! music from South Africa. i was never able to convince myself i knew exactly what the sandinians were doing. there is much italian harmony which is quite complex. i have some how missed the Kung but will look it up. There is so much white influence in south africa, i tend to skip it when looking for traditrional music there, at least ones with interesting scales.
i assume it is all is some form of of a DeBeer temperment:)

Besides, Kraig -- as you point out -- much of the determination of what is harmonic and what is not is in the mind's ear. I was thinking india and the persioan cultures whic hare highly harmonic all best, Johnny Reinhard

p.s. in ancient Greece harmony and melody were reversed!

If anything JI gives a melodic richness that Et do not, while the latter might be argued more useful for harmony.
Lou Harrison found JI his inspiration for melodic material

yes i talked to him about this and we had always agreed on this point,
and as you can see still do

> >

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

🔗Gene Ward Smith <gwsmith@svpal.org>

🔗Dave Keenan <d.keenan@bigpond.net.au>

--- In tuning@yahoogroups.com, "Gene Ward Smith" <gwsmith@s...> wrote:
> --- In tuning@yahoogroups.com, "Dave Keenan" <d.keenan@b...> wrote:
>
> > That means I need to tweak my JI scale definition to say that every
> > note should be reachable from every other note, _and_ the number of
> > just intervals should be greater than the number of notes in the scale
> > (assuming octave-equivalence).
>
> Now if you stack nine of those 19 cycles around a 9 cycle, you get
> 171, which if I am understanding you right you would want to count as
> just intonation. The 19 and 9 cycles themselves have constructions in
> terms of 7-limit consonances, but do not, independently, count as JI
> in spite of the fact that every note is reachable.

Yes. You are understanding me right. 171-ET sounds like JI to me, at
least as 3:5:7-"limit" (max error 0.2 cents), but maybe even the full
9-odd-limit (max error 0.4 cents).

But I'm happy to hear objections.

-- Dave Keenan

🔗Jon Szanto <jszanto@cox.net>

🔗Afmmjr@aol.com

In a message dated 5/25/2005 12:50:56 AM Eastern Standard Time,
kraiggrady@anaphoria.com writes:
i was never able to convince myself i knew exactly what the sandinians were
doing. there is much italian harmony which is quite complex.
There is a famous recording of Sardinians singing in a rich just harmony with
voicies popping out of the harmony...almost every cut on the CD.
i have some how missed the Kung but will look it up.
Kung! are a tribe that tunes up based on the harmonics, relishing the glom of
justness throughout.

If anything JI gives a melodic richness that Et do not
This may be an example of the mind's ear. I do not hear it that way. ET,
with its natural stretches, does more for melodic richness than JI with its
inflexible relationships.

, while the latter might be argued more useful for harmony.
Lou Harrison found JI his inspiration for melodic material

Lou and I spoke about JI as well. First off, he did not believe he should be
quoted all the time with "JI is the best intonation" but acquiessed to this
use by others.

More importantly, Lou introduced extended JI, as with his Simfony in Free
Style and his At The Tomb of Charles Ives (on the Chamber CD on PITCH). This
technique makes greater melodic use of JI than non-modulating JI.
yes i talked to him about this and we had always agreed on this point,
and as you can see still do
Lou also wrote regularly in ET. In fact, it is the tuning for the majority
of his works.

all best, Johnny

🔗Dave Keenan <d.keenan@bigpond.net.au>

I wrote:
> Gene's point is that every note of 19-equal can be reached
> from every other note by a chain of just minor thirds. Change
> "19-equal" to "19 notes of 1/3-comma meantone" if you like, they are
> audibly indistinguishable. It produces the same quandry for my
> definition of a JI scale. If one admits of Pythagorean as being JI,
> then 19-equal is also JI in a similar fairly-uninteresting sort of way.
>
> But I note that the same thing holds for 1/4-comma meantone, since
> every note can be reached from every other note by a chain of just
> major thirds.

This is wrong. 1/4-comma meantone has four parallel chains of just
major thirds, not a single chain, so it is not fully connected by just
intervals at all.

But I'm still inclined to say that for a scale to be called JI
(unqualified), not only should every note be reachable from every
other note by a chain of just (tunable-by-ear) intervals, using only
notes of the scale, but it should also have more such intervals than
there are notes in the scale (assuming octave-equivalence).

Anyone have any good counterexamples on either side of this?

I don't really expect it to be so black-and-white, even assuming we
agreed on which intervals are tunable by ear (which we can't do
because it depends on so many factors).

-- Dave Keenan

🔗Dave Keenan <d.keenan@bigpond.net.au>

I wrote:
> I meant that being a constant structure (CS) was a weaker property
> than being (Rothenberg) proper, not weaker than being MOS or DE.
>
> Rothenberg propriety can be applied to scales with more than two
> interval sizes and there are scales which are CS but are not
> Rothenberg proper e.g. Blackjack. If however there are scales which
> are proper but not CS then I would be wrong to call CS weaker than
> propriety.

Paul Erlich pointed out that there _are_ proper scales which are not
CS, such as 8 notes in a chain of just minor thirds.

So CS is neither a stronger nor weaker condition than propriety.

This usage of "weaker" does not imply any inferiority, only greater
inclusiveness.

Paul also pointed out that all MOS/DE scales are also CS but not
necessarily the other way 'round, so in that sense CS is a weaker
condition than MOS/DE. But of course no-one expects a JI scale to be
MOS or DE.

By the way, I hate limiting the term "MOS" to tunings with only a
single chain of generators per octave.

George Secor recently asked Erv Wilson in a letter, whether he thought
tunings with N parallel chains of generators 1/N of an octave apart
could also be called "linear" and therefore "MOS" when they have only
two step sizes. Unfortunately he did not address the question at all
in his reply.

-- Dave Keenan

🔗Aline Honingh <ahoningh@science.uva.nl>

Dave Keenan wrote:
> > I understand you to be asking, "Given a set of pitches in just
> intonation, what properties determine whether they can be considered a
> scale." Or possibly "a good scale".
> > Some readers thought you were asking what determines whether a scale
> can be considered to be in just intonation.
This is both true, but it seemed useless to mention it afterwards..

> > Tetrachordality is just as much a good property for a JI scale to have
> as a tempered one. In place of equal or linear temperament scales
> being a moment-of-symmetry or being distributionally even, we have the
> property for JI scales of being a "periodicity block". Propriety and
> strict propriety work for JI scales as well as non-JI. A related but
> slightly weaker property that applies to all is that of being a
> "constant structure". Thank you for this answer.

Best regards,
Aline Honingh

🔗Kraig Grady <kraiggrady@anaphoria.com>

While most of Lou's pieces are performed as ET piece he wrote them at his piano tuned to the kirnberger tuning he used for his piano concerto.
He told me
i write all on music on this, and if they want to play it in ET, I can't stop them.
This was his concession to them , which wasn't much for him JI was all about it s melodic qualities, and for myself. That i made this comment in public is what first got Lou talking to me, since he felt the same way. Message: 3 Date: Wed, 25 May 2005 10:03:40 EDT
From: Afmmjr@aol.com
Subject: Re: RE: question about just intonation scales

Lou also wrote regularly in ET. In fact, it is the tuning for the majority of his works.

all best, Johnny

--
Kraig Grady
North American Embassy of Anaphoria Island <http://anaphoria.com/>
The Wandering Medicine Show
KXLU <http://www.kxlu.com/main.html> 88.9 FM Wed 8-9 pm Los Angeles

question about just intonation scales

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