23 Limit Symmetrical Non-Octave Scale

A friend and I have been having some interesting discussion about
Symmetrical Non-Octave Scales, so I thought I post one of mine, in
order to kick off some creative dialog:

23 Limit Symmetrical Non-Octave Scale

Ratio Cents Consecutive
1/1 0.000 0.000
23/22 76.956 76.956
23/21 157.493 80.537
23/20 241.961 84.467
23/19 330.761 88.801
23/18 424.364 93.603
30/23 459.994 35.630
23/16 628.274 168.280
23/15 740.006 111.731
368/225 851.737 111.731
12167/6750 1020.017 168.280
46/25 1055.647 35.630
437/225 1149.250 93.603
92/45 1238.051 88.801
161/75 1322.518 84.467
506/225 1403.055 80.537
529/225 1480.011 76.956

Thanks,

Jacky Ligon

(c) 2001 Jacky Ligon

just joking!

: )

jacky said:

"1/1,23/22,23/21,23/20,23/19,23/18,30/23,23/16,23/15,368/2
25,12167/6750,46/25,437/225,92/45,161/75,506/225,529/225?"

..and I insisted to myself I would not even try this crazy
thing until tommorow but it drove me crazy thinking about
it and so I tried it and I gotta admit Jacky you have got
a real knack for making gorgeous tunings.

Did you have a process you went through to find this one?

- Jeff

> 'wild crazy man' jacky said:
>
> "1/1,23/22,23/21,23/20,23/19,23/18,30/23,23/16,23/15,368/2
> 25,12167/6750,46/25,437/225,92/45,161/75,506/225,529/225?"

OK, I see that you've got this 23 utonal scale pivoting
symmetrically at the 23/15 which is 740 cents and which
you described to me as a 'sweet fifth'.

A sweet fifth at 740 cents!!? I do not know what to
think!! OK, I'll agree it's sweet, but is it a fifth??
Well, maybe it is if I can find some kind of instrument
spectra with a third partial at 1940 cents???

And I've stared at that 30/23 substituting for the 23/17.
I'm sitting around thinking 'how did it occur to him to
do that?' cos I can't think of any way that it would
occur to me to do that. Like, what's the deal with that?
Is it because of folding the pivot point into an octave:
2:1 / 23:15 = 30:23? Is that it? It is a 'secret' octave
reference?

You are driving me crazy Jacky with your tunings that I
cannot figure out and which sound so darn good!!

And furthermore you are completely ruining my
philosophical objections to just intonation! How dare you
send me just scales that I like! What about all of my
years of theorizing and pontificating and developing a
finely crafted bad attitude about JI? All that will have
to be defenestrated now!!

- Jeff

--- In tuning@y..., "J Scott" <xjscott@e...> wrote:

/tuning/topicId_20223.html#20228

> > 'wild crazy man' jacky said:
> >
> > "1/1,23/22,23/21,23/20,23/19,23/18,30/23,23/16,23/15,368/2
> > 25,12167/6750,46/25,437/225,92/45,161/75,506/225,529/225?"
>
> ...
>
> You are driving me crazy Jacky with your tunings that I
> cannot figure out and which sound so darn good!!
>
> And furthermore you are completely ruining my
> philosophical objections to just intonation! How dare you
> send me just scales that I like! What about all of my
> years of theorizing and pontificating and developing a
> finely crafted bad attitude about JI? All that will have
> to be defenestrated now!!

Hah !!

He got you, Jeff !!!!

Go, Jacky, go, go, go !!!!!!!!!!!!

-monz

I gotta admit Jacky you have got
> a real knack for making gorgeous tunings.
>
My point exactly
Mr. Ligon DOES indeed produce very interesting scales.
If he so chooses to share his methods all the better for us but
Do we need to wait for jacky to expire before heeding his greatness?

The present day tuner refuses to die!

Pagano

--- In tuning@y..., "J Scott" <xjscott@e...> wrote:
>
> jacky said:
>
> "1/1,23/22,23/21,23/20,23/19,23/18,30/23,23/16,23/15,368/2
> 25,12167/6750,46/25,437/225,92/45,161/75,506/225,529/225?"
>
> ..and I insisted to myself I would not even try this crazy
> thing until tommorow but it drove me crazy thinking about
> it and so I tried it and I gotta admit Jacky you have got
> a real knack for making gorgeous tunings.

Jeff,

Thanks so much. I do adore 23 - an exotic area along the Harmonic
Mirror sequence.

>
> Did you have a process you went through to find this one?

Yes, see next message.

Jacky Ligon

--- In tuning@y..., "J Scott" <xjscott@e...> wrote:
> > 'wild crazy man' jacky said:
> >
> > "1/1,23/22,23/21,23/20,23/19,23/18,30/23,23/16,23/15,368/2
> > 25,12167/6750,46/25,437/225,92/45,161/75,506/225,529/225?"
>
> OK, I see that you've got this 23 utonal scale pivoting
> symmetrically at the 23/15 which is 740 cents and which
> you described to me as a 'sweet fifth'.
>
> A sweet fifth at 740 cents!!? I do not know what to
> think!! OK, I'll agree it's sweet, but is it a fifth??
> Well, maybe it is if I can find some kind of instrument
> spectra with a third partial at 1940 cents???

Jeff,

Hello! Very often I find myself in the camp of The Dan Stearns
Institute of wacky fifths. Interestingly, Scala doesn't recognize
this as a fifth, but it does function as one in this tuning. I
wonder what is the boundary at which Scala denies "fifthdom" for an
interval. Anyone know? Manuel?

It is interesting to note that Scala recognizes 8 fifths @ an average
of 697.671, but apparently the 23/15 is out of the range of what
Scala considers a fifth.

> And I've stared at that 30/23 substituting for the 23/17.
> I'm sitting around thinking 'how did it occur to him to
> do that?' cos I can't think of any way that it would
> occur to me to do that. Like, what's the deal with that?
> Is it because of folding the pivot point into an octave:
> 2:1 / 23:15 = 30:23? Is that it? It is a 'secret' octave
> reference?

You are correct! This is one species of this kind of scale, where I
wanted to use the inversion of the fifth generator. But as you
perceptively noticed, the other species, does have the 23/17 (utonal
versions exist too). I'm not sure yet which I prefer - I really like
the sound of both.

The interesting and infinitely useful thing about this kind of
construction, is that the symmetry can be "mirrored" from a point
other than the 2/1.

Here I used a short chain of two 23/15. So the 529/225 behaves like
what would be the 2/1 in a "normal" JI tuning. All of the ascending
intervals are merely inverted at the 529/225.

This scale is a constant structure, and an interesting facet of this,
is the "Spiral" of the 23/15 that runs over the range of the keyboard.

> You are driving me crazy Jacky with your tunings that I
> cannot figure out and which sound so darn good!!

Glad to spread a little of this around - I'll go buggy if I keep it
to myself. This kind of tuning is something so near and dear to me,
but I've posted little about it. It is my hope that others will come
forth to speak about their methods of working with non-octave tuning
systems. I'm somewhat familiar with Gary Morrison's 88cet - but I
know there's many more folks too that enjoy these varieties.

> And furthermore you are completely ruining my
> philosophical objections to just intonation!

An interesting statement, as I have probably worked in a reverse
order here - and I know what you mean too. Another huge facet of my
research into this is Non-Octave ETs. This is a wonderful world too,
which has many extremely practical uses. I'll be happy to post about
this too if any are interested.

> How dare you
> send me just scales that I like!

He, he! Thanks! Glad you enjoyed it! : )

> What about all of my
> years of theorizing and pontificating and developing a
> finely crafted bad attitude about JI? All that will have
> to be defenestrated now!!

Well, I think what may turn you off - and it does me too - is "JI
Conservatism". There was a former list member who euphemistically
stated that my goals with rational theory differ dramatically from
most Just Intonationalists. I didn't perceive them to be receptive to
it at all, but I can't disagree with this point either. My music
requires more than 5-7 Limit Ratios. Great rational treasures are
found in the higher primes. It can be a challenging task -
compositionally - to make the leap beyond 5-7, and we see occasional
effort (in surprising places and journals), to make us believe that
nothing of value lies beyond 7. This I categorically reject. And
here's the reason: "5-7" may be good for harmonic languages, which
may care to have music function in a sonic manner, not that different
from 12 tET; but what about the Melodic Frontiers? I have to reject
any philosophy of 5-7 JI which requires me to give up one particle of
the melodic treasures of higher primes. This is the main difference -
I'm not all about harmony and just sounding chord progressions, but
need a broader spectrum of ratios to accommodate the melodic style of
my music. I've found great worth in this area, and have explored up
to the 37 limit to good results. Folks out there in tuning land -
believe your ears, they are the best measure of what's correct for
your music and style.

Then on the flip side, I also enjoy many other types of non-rational
tunings. Shall we begin to discuss Non-Octave ETs?

Thanks for your reply and good humor here - made me laugh out loud!

Jacky Ligon

shreeswifty wrote:
>
> I gotta admit Jacky you have got
> > a real knack for making gorgeous tunings.
> >
> My point exactly
> Mr. Ligon DOES indeed produce very interesting scales.

They look interesting, I just have'nt had
a chance to check 'em out.

--
* 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
* http://mp3.com/davidbeardsley

Any tuning defined as "gorgeous" I'd like to find out about. Please forward
to "Jacky" - sounds like she found one.
Travis
www.freestyleguitar.com

----- Original Message -----
From: "David Beardsley" <xouoxno@virtulink.com>
To: <tuning@yahoogroups.com>
Sent: Saturday, March 17, 2001 5:45 PM
Subject: Re: [tuning] 23 Limit Symmetrical Non-Octave Scale

> shreeswifty wrote:
> >
> > I gotta admit Jacky you have got
> > > a real knack for making gorgeous tunings.
> > >
> > My point exactly
> > Mr. Ligon DOES indeed produce very interesting scales.
>
> They look interesting, I just have'nt had
> a chance to check 'em out.
>
>
> --
> * 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
> * http://mp3.com/davidbeardsley
>
>
> You do not need web access to participate. You may subscribe through
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>

Jacky wrote on 16-3:
>Interestingly, Scala doesn't recognize this (23/15, 740 c.) as a
>fifth, but it does function as one in this tuning. I
>wonder what is the boundary at which Scala denies "fifthdom" for an
>interval. Anyone know? Manuel?

I'm very behind reading the TL. Scala uses the
Blackwood/Rapoport/Regener limits. The lower bound is
2^4/7 = 685.71 cents and the upper bound 2^3/5 = 720 cents.
Outside this range, a 12-tone Pythagorean scale becomes
nonmonotonic, i.e. some pitch classes will get higher tones
than their successors.

Manuel

Manuel wrote,

>Scala uses the
>Blackwood/Rapoport/Regener limits. The lower bound is
>2^4/7 = 685.71 cents and the upper bound 2^3/5 = 720 cents.
>Outside this range, a 12-tone Pythagorean scale becomes
>nonmonotonic, i.e. some pitch classes will get higher tones
>than their successors.

A possible tightening of this definition was proposed by Dave Keenan (
http://www.uq.net.au/~zzdkeena/Music/1ChainOfFifthsTunings.htm
<http://www.uq.net.au/~zzdkeena/Music/1ChainOfFifthsTunings.htm> ):

"From 714.9 cents up to 720 cents (5-tET), a chain of 6 fifths produces a
better 2:3 than the fifths making up the chain. Similarly from 689.3 cents
down to 685.7 cents (7-tET), a chain of 8 fifths produces a better 2:3 than
the fifths making up the chain.This provides a handy functional definition
of a wolf fifth for our purposes, namely any fifth for which some chain of
less than 11 (*) gives a better approximation of 2:3. This corresponds to
any fifth outside the range 689.3 to 714.9 cents, i.e. any fifth with an
error outside -12.7 to +12.9 cents."

--- In tuning@y..., <manuel.op.de.coul@e...> wrote:
>
> Jacky wrote on 16-3:
> >Interestingly, Scala doesn't recognize this (23/15, 740 c.) as a
> >fifth, but it does function as one in this tuning. I
> >wonder what is the boundary at which Scala denies "fifthdom" for an
> >interval. Anyone know? Manuel?
>
> I'm very behind reading the TL. Scala uses the
> Blackwood/Rapoport/Regener limits. The lower bound is
> 2^4/7 = 685.71 cents and the upper bound 2^3/5 = 720 cents.
> Outside this range, a 12-tone Pythagorean scale becomes
> nonmonotonic, i.e. some pitch classes will get higher tones
> than their successors.
>
> Manuel

Manuel and other scholars,

In a private dialog with another list member, the very fascinating
question came up about: "What exactly is an octave?". In other words,
how do we define the boundaries of the octave (if we dare)? And here
I'm talking about the "perceptual octave" that is either stretched or
compressed in the music of multitudes of cultures around the world;
some preferring it wide, some liking it small. Here the exact
mathematical 2/1 of JI and EDOs is now where to be found, and the
music sounds beautiful, and is heard within the cultural context as a
desirable sound in the music.

The pertinent question being: If the Blackwood/Rapoport/Regener
limits allow that 3/2 may be between 2^4/7 = 685.71 cents and the
upper bound 2^3/5 = 720 cents, then may we use a similar measure to
define a similar boundary for our beloved 2/1?

Or do we hold it to be a "too sacred object" to follow the model of
large portions of the world in constructing scales, and insist on
keeping it "un-violated"?

I believe this is somewhat dependent upon the timbres being used, and
the musical context, where melody and rhythmic motion come into play.

IMHO (and the HO of many I know), this is a great wellspring of
tuning possibilities, nearly untapped where we insist on the
supremacy of an exact 2/1 ratio. My experience, along with many
others who have explored these realms, is that there *is* a band
around the 2/1, which when used in the context of music, is
functionally and perceptually the same as a 2/1, but may have an
energetic and slight beating which is quite lovely. I believe we can
easily look at this as a flexible interval, in the same manner as we
have came to regard the 3/2 as flexible (I should note that on my
piece "Ten Thousand Things" there is singing with a 750 cents "fifth"
which does not shock the ear, but sounds wonderfully energetic,
displacing even here, what we may call a 3/2), without committing any
violation to the sound of tunings, scales and music. We only gain
from this view a boundless freedom. And when we arrive at the doors
of music with all this, we find things are quite a bit more flexible
than we ever imagined while looking through a more theoretical
conception of tunings. At music's door theory becomes realized and
sometimes discarded for the realities of what the ear, and our
emotions reveal.

Any thoughts?

Thanks,

Jacky Ligon

According to my harmonic entropy model (with "typical" resolution of
1% assumed), any interval in the range

667.1¢ to 736.7¢

is more likely to be heard as a 3:2 ratio than to be heard as any
other ratio.

Similarly, any interval in the range

1158.0¢ to 1242.0¢

is more likely to be heard as a 2:1 ratio than to be heard as any
other ratio.

--- In tuning@y..., paul@s... wrote:
> According to my harmonic entropy model (with "typical" resolution
of
> 1% assumed), any interval in the range
>
> 667.1¢ to 736.7¢
>
> is more likely to be heard as a 3:2 ratio than to be heard as any
> other ratio.
>
> Similarly, any interval in the range
>
> 1158.0¢ to 1242.0¢
>
> is more likely to be heard as a 2:1 ratio than to be heard as any
> other ratio.

Paul,

This does show quite a bit flexibility, and for the area of the 3/2
your findings seem to accord well with the Blackwood/Rapoport/Regener
limits. And this is a nice judicious band around the 2/1 as well. I
could in fact see this possibility looking over your harmonic entropy
graph.

May I ask how you arrive at the ""typical" resolution of 1%"?

Jacky

--- In tuning@y..., ligonj@n... wrote:

> Paul,
>
> This does show quite a bit flexibility, and for the area of the 3/2
> your findings seem to accord well with the
Blackwood/Rapoport/Regener
> limits.

My range actually shows more flexibility, allowing for Pelog and
Slendro MOS scales (as Wilson saw them).

> And this is a nice judicious band around the 2/1 as well. I
> could in fact see this possibility looking over your harmonic
entropy
> graph.

You do find octaves mistuned by almost a quartertone in many cultures.
>
> May I ask how you arrive at the ""typical" resolution of 1%"?

It's the one arbitrary parameter of the model. For sine waves,
Goldstein found that different listeners have a resolution (in this
context) of 0.6% to 1.2% in an ideal frequency range, around 3000Hz.
If we are using instruments with partials in this range, 1% seems like
a good round number to use. For further discussion, and if you wish to
reply to me, I refer you to the harmonic entropy list
(harmonic_entropy@yahoogroups.com).

Paul Erlich wrote:

> According to my harmonic entropy model (with "typical" resolution of
> 1% assumed), any interval in the range
>
> 667.1� to 736.7�
>
> is more likely to be heard as a 3:2 ratio than to be heard as any
> other ratio.
>
> Similarly, any interval in the range
>
> 1158.0� to 1242.0�
>
> is more likely to be heard as a 2:1 ratio than to be heard as any
> other ratio.

The extremes of those ranges would sound horribly mistuned using instruments
with harmonic partials. Might work with bells, or a in a Sethares type of
context.

Here's another monkey that could be thrown into that wrench. What happens
when you have two scale degrees within one of those ranges, one toward the
middle and one toward the outside? Say you have a scale with degrees at both
694� and 736�. Or you stretch your octave to 1242� and end up with a
"leading tone" of 1195�. [In the voice of the Godfather:] What then? Huh?

Sticky hypothetical situations aside, I would prefer to define the octave in terms of
Western music theory. I would use a different term for approximations of the 2:1 in
other musics. The word "octave" has the Latin word for "eight" in it, implying a
heptatonic scale.

Jacky,
Using stretched and compressed 2:1 to achieve greater consonances in ETs has
been discussed at length here before, maybe about 2 years ago, I'm thinking. I don't
know whether the e-Groups/Yahoo archives go back that far or not but it might be
worth while to do a search.

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

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

> Here's another monkey that could be thrown into that wrench. What happen=
s
> when you have two scale degrees within one of those ranges, one toward th=
e
> middle and one toward the outside? Say you have a scale with degrees at =
both
> 694¢ and 736¢. Or you stretch your octave to 1242¢ and end up with a
> "leading tone" of 1195¢.

Hard to see how a more accurate 2:1 could possibly "lead" to a less accurat=
e 2:1.

>[In the voice of the Godfather:] What then? Huh?

You still don't hear 736 cents or 1242 cents as a definite frequency ratio =
-- each just
sound like an out-of-tune version of its lower neighbor in the scales you'v=
e posited.
>
> Sticky hypothetical situations aside, I would prefer to define the octave=
in terms of
> Western music theory. I would use a different term for approximations of=
the 2:1 in
> other musics. The word "octave" has the Latin word for "eight" in it, im=
plying a
> heptatonic scale.

Well in that case I think Jacky and I are really talking about a different =
concept. Neither
one of us want to be restricted to using only heptatonic scales.

--- In tuning@y..., paul@s... wrote:
> You still don't hear 736 cents or 1242 cents as a definite
frequency ratio -- each just
> sound like an out-of-tune version of its lower neighbor in the
scales you've posited.

This is where I must depart from thee!! To say that things sound out
of tune relative to this or that, is to imply that there are polar
opposites. And here I must plug in the famous words of Ivor Darreg:

"THERE ARE NO BAD SCALES"

Now my view on this:

I want to tear off my cloths and run through the meadows like I'm in
a microtonal nudist colony, where nothing is concealed and everything
is revealed in the name of music! I refuse to deny myself the broader
reality of tunings outside 5-7 limit JI. My ears grew to need this
over ten years ago - and I *don't* think 5-7 represents everything
that is valid about scales. How about we think of something other
than "out of tune" relative to our conceptions (and preconceptions)
of what 5-7 limit has to offer? How about the wonders of 11, 13, 17,
19, 23 (one of my favorites), and beyond? Now don't get me wrong - I
*know* the beauty of 5-7 *intimately*, from years of using it in
actual music compositions - but I can't let my mind become frozen to
other *equally* beautiful realities. Surely there are better ways to
say things, that will not imply that things beyond 5-7 Limit Ratios
are "out of tune" - where these stretched or compressed intervals
that fit within and neighbor these lower members of the harmonic
series; and thereby, imply by our use of terms, that they may be
invalid. There are myriad of other consonances and near consonances
to be cherished, and should be considered treasures for music
creation, every bit on the level of 5-7 limit JI. I suggest other
metaphors, even though I don't know what they might be right now. All
I know, is that if we but forth this to those new to tuning theory,
and who may hope to make music with it, then they may have an
unnecessarily warped view early on in the course of their learning.

Beauty is what we hear folks - never discount this. And yes - beauty
is subjective - and this is exactly why it's important to your music
with microtonality; you decide with your ears what sounds right, then
you'll know the most important reality - *your reality*. Your reality
is what has made centuries of music sound beautiful, and will for
centuries to come. "Out of tune" is meaningless, because it implies
good scales and bad scales. I like beating (no not my chest!) - huge
portions of the world like beating. Having beatless intervals is kind
of one dimensional anyhow (a laboratory construct based in
mathematics) - and really a myth of sorts, because no matter what you
do - now get yer faces close to that monitor out there on this one:
you are going to have beating in music where you have overlapping
harmony and melody no matter how MIRACULOUS your scales are. Beating
makes music live. I could never say this if I didn't have 15 years of
experience wallowing in this reality almost daily, and from the
effort, have enough music to set you down for a week and play
continuous original microtonal music.

> Well in that case I think Jacky and I are really talking about a
different
> concept. Neither one of us want to be restricted to using only
heptatonic scales.

Right - nor do I wish to be constricted by conceptions of "out of
tune" relative to this and that. Freedom - Freedom - FREEDOM!! I toss
all these phrases out the window, so I can get on with tuning up my
synths and samplers, and getting wonderful friends to sing in
microtonal compositions without 3/2 and 2/1 or 5-7 limit JI anywhere
to be found.

Tenderly,

Jacky Ligon

In 1931 the mathematician and logician Kurt Godel proved that within
a formal system questions exist that are neither provable nor
disprovable on the basis of the axioms that define the system. This
is known as Godel's Undecidability Theorem. He also showed that in a
sufficiently rich formal system in which decidability of all
questions is required, there will be contradictory statements. This
is known as his Incompleteness Theorem.

In establishing these theorems Godel showed that there are problems
that cannot be solved by any set of rules or procedures; instead for
these problems one must always extend the set of axioms. This
disproved a common belief at the time that the different branches of
mathematics could be integrated and placed on a single logical
foundation.

Paul E. wrote:

> Or you stretch your octave to 1242� and end up with a
> > "leading tone" of 1195�.
>
> Hard to see how a more accurate 2:1 could possibly "lead" to a less accurat=
>
> e 2:1.

Exactly my point. So the 1242� interval "becomes" the first chromatic semitone in the next octave (if the
scale is being used in anything like a Western way). It's not going to sound like an approximation to the
octave if it's used melodically next to it's nearest neighbor.

> >[In the voice of the Godfather:] What then? Huh?
>
> You still don't hear 736 cents or 1242 cents as a definite frequency ratio =
>
> -- each just
> sound like an out-of-tune version of its lower neighbor in the scales you'v=
>
> e posited.

Well, I see I'm going to have to quit positing and start playing to find out the answer to my (rhetorical)
question.

> I would prefer to define the octave=
>
> in terms of
> > Western music theory. I would use a different term for approximations of=
>
> the 2:1 in
> > other musics. The word "octave" has the Latin word for "eight" in it, im=
>
> plying a
> > heptatonic scale.
>
> Well in that case I think Jacky and I are really talking about a different =
>
> concept. Neither
> one of us want to be restricted to using only heptatonic scales.

Neither do I. And I'll let Jacky speak for himself on that one, if he wishes (I have no doubts! :-). It
seems to me that this question (in the Subject line), in practice, comes down to "Is 2:1 or there abouts an
unavoidable point of perceptual equivalence or not?" I think it is. But I also think that context plays a
huge role in how far it can be stretched. Harmonic entropy is tremendously useful but it only goes so far
(MHO).

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

--- In tuning@y..., ligonj@n... wrote:

> This is where I must depart from thee!! To say that things sound out
> of tune relative to this or that, is to imply that there are polar
> opposites. And here I must plug in the famous words of Ivor Darreg:
>
> "THERE ARE NO BAD SCALES"

You must have completely misunderstood me, Jacky! I think a
scale with several different approximations to a simple ratio could
be a really GOOD scale, with lots of great expressive potential.

--- In tuning@y..., paul@s... wrote:
> --- In tuning@y..., ligonj@n... wrote:
>
> > This is where I must depart from thee!! To say that things sound
out
> > of tune relative to this or that, is to imply that there are
polar
> > opposites. And here I must plug in the famous words of Ivor
Darreg:
> >
> > "THERE ARE NO BAD SCALES"
>
> You must have completely misunderstood me, Jacky! I think a
> scale with several different approximations to a simple ratio could
> be a really GOOD scale, with lots of great expressive potential.

Paul,

Oops! Sorry. It's just the oft used "out of tune" metaphor that
carries mysterious connotation for me, and which prompted my remarks.
Thanks for the clarification. I guess I can put my clothes back on
now.

JL

--- In tuning@y..., ligonj@n... wrote:

/tuning/topicId_20223.html#22159

> I want to tear off my cloths and run through the meadows like I'm
in a microtonal nudist colony, where nothing is concealed and
everything is revealed in the name of music!

This list does get "exciting" from time to time, but I must admit
that this is one of the more "piquant" images...

>
> In 1931 the mathematician and logician Kurt Godel proved that
within a formal system questions exist that are neither provable nor
> disprovable on the basis of the axioms that define the system. This
> is known as Godel's Undecidability Theorem.

This is excellent news! I wasn't quite sure which interval to use,
either!....

_________ ____ ______ _____
Joseph Pehrson