Tuning-Math Digests messages 5826 - 5850

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Message: 5826

Date: Sat, 04 Jan 2003 08:21:28

Subject: Re: Fwd: Re: A common notation for JI and ETss

From: David C Keenan

George Secor:

>Dave Keenan:
> > I'd prefer to go with )|( as the 7:11 comma since it only involves
>a 0.55
> > cent schisma. I feel that a 3 flag symbol for something under 10
>cents
> > could not be justified when a 2 flagger is within 0.98 cents.
>
>Agreed, but I would call it the 7':11 comma for reasons given below.
>
> > It seems 891:896 )|( should be called the 7:11 comma while the
>comma
> > represented by (| is called the 7:11'-comma.
>
>Before we used the colon designation for these two-prime commas, we
>were expressing them as the sum or difference of two single-prime
>commas, e.g., the 5:7 comma was the 7-5 comma.  How would you do that
>for 891:896 other than as the 11'-7' comma?  (However, since you
>don't like what I have for the 7' comma, see below.)  Since this is
>the comma that is arrived at by invoking a new (7') comma, I think
>that this should be called the 7':11 comma.

Why not 7':11' if it is 11' - 7' ? But see my alternative suggestion below.

> > Are there any ETs in which we should now prefer )|( over some other
>symbol
> > given that it now has such a low prime-limit or low product
>complexity?
> >

I'll just note that neither of us have answered the above yet, in case the 
way I edited things might have made it look like the following was 
answering it, which of course it is not.

> > >They are all 7-related.  In a 13-limit heptad (8:9:10:11:12:13:14)
>it
> > >is 7 that introduces scale impropriety; e.g., the fifth 5:7 is
>smaller
> > >than the fourth 7:10.  Replace 14 with 15 in the heptad and I
>believe
> > >the scale is proper.  So it would not be surprising that someone
>might
> > >want to respell the intervals involving 7 -- 4:7 as a sixth, 5:7
>as a
> > >fourth, 6:7 as a second, 7:9 as a fourth, 11:14 as a third, and
>13:14
> > >as an altered unison.
> > >
> > >So we would want to notate the following ratios of 7 using these
> > >commas:
> > >
> > >                              deg217   deg494
> > >                              ------   ------
> > >A# 32768:59049  ~1019.550c     185      420
> > >vs. 7/4          ~968.826c     175      399
> > >57344:59049       ~50.724c      10       21
> > >(apotome complement of 27:28 - this could be called the 7' comma)
> > >11:19 comma (|~   ~49.895c       9       21
> > >But a new symbol /|)` would represent it exactly
> > >(if the flags are added up separately ­ 5+7+5' comma)
> >
> > I really don't think it is necessary or desirable to notate this 7'-
>comma.
> > It is larger than the standard 7-comma and it involves a longer
>chain of
> > fifths than _any_ other comma we've ever used.
>
>I think it's a matter of waiting to see if we'll have to, because I
>don't think that wanting to notate 7/4 relative to C as A-something
>would be unusual or weird.

But this one is notating it as A#-something, not A-something. Sure in 
meantone you can notate 4:7 as C:A# but you'd only do it because the 
something happens to vanish, it's a long way up the chain of fifths and Bb 
\!/ is likely to be more convenient.

> > I think we should only accept the need for a _larger_ alternative
>comma for
> > some prime (or ratio of primes) if it involves a _shorter_ chain of
>fifths.
> >
> > >Expressed another way:
> >
> > I don't see the following quote as expressing the above quote
>another way.
> > It is a completely different 7-comma. With this comma a 4:7 above C
>would
> > be a kind of A, not A#.
>
>Okay, then call its apotome complement, 27:28, the 7' comma and use
>the 13'-5' symbol (|\' to represent it (replacing ' with whatever we
>eventually agree on for the 5' comma).  I notice that this is the
>next symbol I proposed:
>
> > A  16:27
> > vs. 4:7
> >
> > >F  3:4           ~498.045c      90      205
> > >vs. 9/7          ~435.084c      79      179
> > >27:28             ~62.961c      11       26
> > >symbol  )||                     12       26
> > >But a new symbol (|\' would represent it exactly
> >
> > It is very large,
>
>But it's still smaller than the 13' comma, ~65.3c, so this doesn't
>take us outside our upper boundary for single-shaft symbols.

True.

> > and the absolute value of its power of 3 is still larger
> > than that of the standard 7 comma, although only by 1. I'm not
>convinced
> > there's any need for it.
>
>As I said, let's wait and see.  The nice thing about this is that it
>doesn't require any new flags other than the 5' (for which it offers
>further justification for having that new flag or whatever) and that
>it's exact.  Come to think of it, I seem to recall that Margo wrote
>me a couple of weeks ago that she wanted a JI symbol for 27:28 -- you
>must admit that this is not a weird or unusual interval.

Certainly not an unusual interval, and we can already notate it. 27:28 from 
C is of course Db!). But I understand you mean a symbol that represents it 
as a modified unison. I can see that this might be useful, although I don't 
think its apotome complement will be of much use.

>   If there is
>any problem with this, I think it is that we need to be able to
>represent the 5' comma in such a way that the symbols in which it is
>used don't look weird.

That would be nice, but I would still want to avoid its use as much as 
possible.


> > >F#  512:729      ~611.730c     111      252
> > >vs. 7/5          ~582.512c     105      240
> > >3584:3645         ~29.218c       6       12
> > >This is the 5:7' comma, or 7+5' comma, or 7'-5 comma
> > >A new symbol  |)` would represent it exactly
> >
> > This contains 3^6 while the standard 5:7-comma has 3^-6 so I think
> > there  could be some demand for this one. I think the proposed
>symbol is
> > good, being only 2 flags, however I'd like it even better if we
>could come
> > up with some way that the 5'-comma (ordinary schisma) could be
>notated as a
> > modification of the shaft rather than as a flag, or if the two
>flags were
> > not on the same side.
>
>We need to find a good way to represent *both* the 5' and -5'
>alterations that involves something other than a flag -- something
>laterally aligned with the shaft, if not a modification to the shaft
>itself.  (So back to the drawing board!)

Agreed.

> > But in any case, it seems we should avoid using it if possible
>because of
> > its containing that very unfamiliar flag. It's kind of strange if
>we should
> > need to use this obscurte new flag as low as the 7-limit. You
>should leave
> > it out of the XH18 paper.
>
>I have a feeling that the 5' comma is going to be useful for notating
>all sorts of things regardless of the prime limit (we have already
>proposed incorporating it into the diaschisma, Pythagorean comma, and
>5-diesis symbols), particularly if it will indicate intervals
>exactly, so I wouldn't call this an obscure flag -- just a very small
>one.  And the idea of using something other than a lateral flag to
>symbolize it strikes me as highly appropriate -- just so long as it
>looks good (and therein lies the problem).
>
> > >E  64:81         ~407.820c      74      168
> > >vs. 14/11        ~417.508c      75      172
> > >891:896            ~9.688c       1        4
> > >5:7+19 comma )|(   ~9.136c       2        3
> >
> > Agreed.
> >
> > >C#  2048:2187    ~113.685c      21       47
> > >vs. 14/13        ~128.298c      23       53
> > >28431:28672       ~14.613c       2        6
> > >17' comma ~|(     ~14.730c       3        6
> >
> > Agreed. I though we already had that one. I believe we called this
>the 7:13
> > comma while (|( is the 7:13' comma.
>
>I have been calling (|( the 7:13 comma, since it is the 13'-7 comma;
>however 28431:28672 isn't the 13-7 comma -- it's the 7'-13 comma (if
>27:28 is now the 7' comma), so I would then also call it the 7':13
>comma.
>
>If 27:28 is the 7' comma, then I would also have to rename the
>following in what I gave above:
>
>3584:3645 as the 5:7' comma or 7+5' comma (but now not the 7'-5 comma)
>891:896 as the 7':11 comma or 7'-11 comma

When naming these commas, how about we forget about the details of whether 
an x:y comma is the sum or difference of x or x' and y or y' but simply 
parse "x:y'-comma" as "(x:y)'-comma" meaning simply the second (and less 
important) x:y comma.

I noticed recently, an ambiguity in the spoken form of these comma names. 
To say "x prime comma" can be taken as merely referring to the fact that x 
is a prime number. Perhaps in the spoken form it would be more useful to 
say "small x comma" or "big x comma", except that this doesn't correspond 
directly to primed or unprimed because we have the 5'-comma smaller than 
the 5-comma while all the others have primed larger than unprimed.

We could refer to the unprimed one without using "big" or "small" and use 
whichever of these applies for the primed one. If really necessary the 
unprimed could be called "normal" or "standard".

>So have I sold you on a 7' comma, 27:28?

Yes. It's acceptable because it only contains 3^3. I feel there is some 
sort of comma uselessness metric that increases with both the size of the 
comma and the number of fifths. A first guess would be to take the product 
of these. I wouldn't want to add any comma to the system that had this 
"uselessness" higher than any we've already got. The highest so far is 362 
for the 11' comma (|), and a close second is the 25 comma //| at 344.

The proposed 7' comma (27:28) is ok at 189, but 57344:59049 is out of the 
question at 507.

I think we should also dump the 31" comma (65536:67797) at a uselessness of 
411.

Actually, I think the number of fifths should feature more strongly in 
uselessness, or we could have a sharp cutoff at 9 fifths.

But I'm still reluctant re 27:28 because of the added complication of the 
5'-comma "flag". No matter how good we can make it look, the fact remains 
that we haven't had to use it at all for anything else in the 15-limit plus 
harmonics to 31. So I think it should not be considered part of the "basic" 
system despite its low prime limit and low product complexity.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 5828

Date: Sat, 04 Jan 2003 00:09:44

Subject: Fwd: Re: Temperament notationn

From: Dave Keenan

> Any other ideas that might favor 50?

No, but it might make sense to say that for meantones that are on the
1/3-comma side of golden meantone (or some such boundary) a
50-ET-based notation may be preferable.


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Message: 5829

Date: Sat, 04 Jan 2003 01:16:55

Subject: Re: Temperament notation

From: Carl Lumma

>>That was a bad choice of terminology.  The issue that Gene and
>>I are raising is: should notation be based on the melodic scale
>>being used, or should they be based on the 7-tone meantone
>>diatonic scale musicians are already familiar with?
>
>The issue that I raise is: how many different notations can you 
>expect a person to learn?

As many as he needs to play the music he wants to.

Since learning septimal notation gives access to 300 years of
Western music, the potential payoff for each new one shouldn't
be grounds for complaint.

>>>But what value is the decimal notation to me, and what incentive
>>>would I have to learn it without a decimal keyboard?
>> 
>> () It gives you an invaluable tool for understanding the music.
>
>Okay, but only as long as the music is written in the Miracle 
>temperament, yes?  (More about this below.)

Only if the music is written in a decatonic scale, yes.  There
are many ways to supply the chromatic pitches outside of Miracle
temperament.

>The tones would be on a diagonal row of keys that (ascending in 
>pitch) would go off the near edge of the keyboard; but they could
>be picked up at the far edge, so yes, it can be done without
>extraordinary effort.

I'd say there's nothing far worse about such a setup than in
using the Halberstadt for 12-equal.
 
>>>Now give me the same decimal keyboard with Partch's 11-limit
>>>JI mapped onto it (observe that this was the reason that I
>>>originally came up with the layout).  Again, I should do just
>>>fine with 72-ET sagittal notation, assuming that I am proficient
>>>with the keyboard.
>>
>>You seem to be saying that it's easier to learn to find pitches
>>on a keyboard than it is to learn to find pitches in a notation...
>>For me, it's the opposite.
> 
>Actually I do find it easier to perceive the pitch relationships
>on a generalized keyboard (of whatever sort) than from a notation,

Well, that's an important statement.  Since I don't have any
experience playing a generalized keyboard of any sort, I have
nothing to offer.  I can say that learning to read music was
easier for me than learning to finger the piano.  I really have
no idea if this relationship would remain when learning a new
keyboard/notation pair.

>The broader point that I was trying to make seems to have gotten
>lost in all of the details of the discussion.  I was trying to
>show that there is no particular advantage in using decimal
>notation to notate music that is *not* based on the Miracle
>geometry (e.g., Partch's music, which is better understood in
>reference to an 11-limit tonality diamond), but for which the
>tones may still be very suitably mapped onto a decimal keyboard.
>The advantage of decimal notation comes into effect only when and
>if you are using the Miracle temperament itself, i.e., exploiting
>the tonal relationships that are unique to Miracle.  Likewise, if
>I play something in 31, 41, or 72-ET on a decimal keyboard that
>was composed by someone utterly ignorant of Miracle as an
>organizing principle for tonality (as I believe *all* of us were
>up until a couple of years ago -- myself included), is the decimal
>notation going to benefit me in any way if the composition which
>I am playing was not conceived as being decatonic?

No!  But there are other temperaments with interesting decatonic
scales besides 31, 41, and 72, and all of them would get ten
nominals under my pen, and I suspect the differences in
accidentals would be easy for performers to learn.

>But it would be unrealistic to expect anyone to learn three
>different notations for one tonal system, according to which
>tonal relationships are exploited in a given piece.

Perhaps we'll have to agree to disagree.

>(Or suppose that a piece is heptatonic in one place and decatonic
>in another.  Do we switch notations in the middle of the page?)

Absolutely!  Just like switching clefs or key signatures.

>My point is that alternate tunings often do not tie us down to 
>specific tonal organizations, so the choice of a tuning is often
>not enough to determine how many nominals would be "best" for its
>notation.

Indeed, any time we leave "diatonic" writing, the need to show
scale intervals in the notation disappears.  Then what I meant
by 'transpositionally invariant' notation would be optimal -- a
notation in which acoustic intervals always look the same.  In
diatonic notation it is scale intervals (2nds, 3rds, etc.) that
always look the same.

You mention Partch's music, which doesn't really use any fixed
melodic scale.  I would think transpositionally invariant
notation would be optimal, for the scores at least.

But Partch had the right idea... since his instruments played
different scales (tonality diamond, microchromatic scales,
ancient melodic scales, etc.), he notated differently for each
of them.

>Since we are already acquainted with a notation that uses 7
>nominals, and if that works reasonably well for many alternative
>tunings, then why not have a generalized notation that builds on
>that?

I am genuinely interested to see how it looks.  You should debut
it with sample music, both original and classical.
 
>So I would therefore require a microtonal musician to learn no
>more than one new notation.  A composer may wish to do otherwise
>when composing, but a translation would be provided for the player.

A distinct posibility.

>>Let's ask it this way: take the well-tempered clavier and re-
>>write it with 6 nominals.  Is that only a slight disadvantage?
>
>I was about to say no, but only because 6 nominals will hardly
>work well with anything,

They'd work spledidly for wholetone music.  It's fortunate that
12-equal supports the wholetone scale without collisions.  But
if you look at octatonic music, it would be much better notated
with 8 nominals.  I would go so far as to suggest that this held
back octatonic music in the last century.

>even for Partch (since 6:7:8:9:10:11 isn't a constant structure).

The diatonic scale in 12-equal isn't a constant structure either.
Actually, it's strict propriety that's important, and the
violations aren't too bad here, and I think 6 nominals would be
ideal.  But Partch doesn't really stick to this scale, so...

>I'm not really arguing against specialized notations with other
>than 7 nominals, but I don't think that we can expect very many
>players to learn them.

They'll learn them if there's cool music written in them.

Learning any notation is a tremendously difficult problem
for humans, but they never cease to amaze me.

>I have no doubt that the decimal keyboard and notation together
>would be easy and even fun to learn, but I wonder whether very
>many persons would ever have the opportunity to do it.

Did you see any of the virtual keyboard projector posts I've
made to the main list over the past year?

-Carl


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Message: 5832

Date: Sat, 04 Jan 2003 19:27:43

Subject: Re: Temperament notation

From: David C Keenan

At 12:12 AM 4/01/2003 +0000, Dave Keenan <d.keenan@xx.xxx.xx> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx "Gene Ward Smith
><genewardsmith@j...>" <genewardsmith@j...> wrote:
>--- In tuning-math@xxxxxxxxxxx.xxxx "gdsecor <gdsecor@y...>"
><gdsecor@y...> wrote:
>
> > Since the 5-limit optimal generator size depends on the optimizing
> > method, then it's a tossup between 31 and 50.
>
>One rather rough and ready rule along these lines would be to pick the
>last division which occurs as a denominator of a convergent for both
>the rms and minimax generator.

That sounds like an excellent idea!

>For Miracle, we get 72 in both the 7 and 11 limits, and for meantone,
>we get 31 in both the 5 and 7 limits (11 limit too, obviously; I
>didn't bother to compute it since it was clear what the result would
>be.) For the 7-limit schismic, we have 94, but for 5-limit, it runs
>all the way up to 289. I suppose 53, 118, 171 and 289, all very
>similar from a 5-limit schismic point of view, would look completely
>different when notated sagitally?

As a matter of fact there is a sagittal notation that agrees with 118, 171 
and 289 and is also compatible with 53 and 94. In the two-symbol form, it 
strongly resembles a linear-temperament-specific notation with 12 nominals, 
like the decimal notation for miracle. This is possible because the 
generator happens to be a fifth and the chroma happens to be the 5-comma 
(80;81).

A chain of 60 notes looks like this.

Eb\\! Bb\\! F\\! C\\! G\\! D\\! A\\! E\\! B\\! F#\\! C#\\! G#\\!
Eb\!  Bb\!  F\!  C\!  G\!  D\!  A\!  E\!  B\!  F#\!  C#\!  G#\!
Eb    Bb    F    C    G    D    A    E    B    F#    C#    G#
Eb/|  Bb/|  F/|  C/|  G/|  D/|  A/|  E/|  B/|  F#/|  C#/|  G#/|
Eb//| Bb//| F//| C//| G//| D//| A//| E//| B//| F#//| C#//| G#//|

The reason I say it's only "compatible" with 53 and 94 is because the 
standard sets for these do not use the 25-comma symbol //|

I suggest this notation (or its single-symbol counterpart) should be used 
for all open schismics of up to 60 notes.

-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 5833

Date: Sun, 05 Jan 2003 08:31:25

Subject: Re: Temperament notation

From: David C Keenan

Notice that this sagittal notation for schismic depends only on the 
temperament's mapping from generators to primes and not on any particular 
range of generator sizes. The fact that it only uses 5-limit comma symbols 
(if //| is taken as a double 5-comma rather than a 25 comma) means that it 
is valid for schismic at all odd limits.

Eb\\! Bb\\! F\\! C\\! G\\! D\\! A\\! E\\! B\\! F#\\! C#\\! G#\\!
Eb\!  Bb\!  F\!  C\!  G\!  D\!  A\!  E\!  B\!  F#\!  C#\!  G#\!
Eb    Bb    F    C    G    D    A    E    B    F#    C#    G#
Eb/|  Bb/|  F/|  C/|  G/|  D/|  A/|  E/|  B/|  F#/|  C#/|  G#/|
Eb//| Bb//| F//| C//| G//| D//| A//| E//| B//| F#//| C#//| G#//|

This points the way to similar multi-symbol sagittal notations for other 
linear temperaments (LTs). As well as being as independent of generator 
size as possible and hence independent of any particular ET (which may not 
be fully acheivable when the generator is not a fifth) it will make the 
sagittal notation for an LT resemble as closely as possible the ideal 
notation where more than 7 nominals are allowed.

You first decide how many nominals there should be. Call this N. Then 
examine the available symbol commas in order of popularity, applying the 
LT's primes-to-generators mapping, until a symbol is found that corresponds 
to a chain of N generators (the chroma), also possibly 2N, 3N etc.

Naming the notes of the central chain is another matter which I haven't 
worked out in general, except by using an ET.

For example, in the case of Miracle, we would want 10 nominals and we find 
that |) as the 7-comma 63;64 corresponds to the chroma.

C     C#/|  D|)   Eb/|\ E||)  F||\  G     G#/|  A|)   Bb/|\
C!)   C#\!  D     Eb|\  E|)   F/|\  G!)   G#\!  A     Bb/|
C!!)  C#\!/ D!)   Eb!/  E     F/|   G!!)  G#\!/ A!)   Bb\!

The following is perhaps a more natural sagittal notation (based on 72-ET), 
but makes no attempt to look like it has 10 nominals.

C     Db/|  D|)   E\!/  F!)   F#\!  G     Ab/|  A|)   B\!/
C!)   C#\!  D     Eb|\  E|)   F/|\  G!)   G#\!  A     Bb/|
B|)   C/|\  D!)   D#!/  E     F/|   Gb|)  G/|\  A!)   A#\!
B
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 5835

Date: Sun, 05 Jan 2003 08:40:46

Subject: Re: Temperament notation

From: David C Keenan

At 02:32 AM 4/01/2003 +0000, wallyesterpaulrus 
<wallyesterpaulrus@xxxxx.xxx> wrote:
>dave,
>
>MAD means "Mean Absolute Deviation"
>
>which is another error criterion yet.
>
>so MA would breed confusion.

OK. I admit defeat on the acronym. But I'd still prefer to call it 
max-absolute rather than minimax.

>--- In tuning-math@xxxxxxxxxxx.xxxx "Dave Keenan <d.keenan@u...>"
><d.keenan@u...> wrote:
> > --- In tuning-math@xxxxxxxxxxx.xxxx "Paul G Hjelmstad"
> > <paul.hjelmstad@u...> wrote:
> > >
> > > Would someone explain "minimax" generator (I understand rms
>generator)
> > >
> > > Thanks
> >
> > Frankly I think "minimax" is a silly term. I prefer to call it the
> > max-absolute (MA) generator. Obviously we're trying to minimise the
> > error measure in both cases, but we don't say "miniRMS". In one case
> > we minimise (I'd prefer to say "optimise") the Root of the Mean of
>the
> > Squares of the errors and in the other it is the Maximum of the
> > Absolute-values of the errors. RMS and MA.

-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 5836

Date: Sun, 05 Jan 2003 15:42:45

Subject: Re: A common notation for JI and ETs

From: David C Keenan

Here's my latest suggestion regarding symbolising the 5'-comma (5-schisma) 
up and down:

Make them separate symbols. Like an accent mark on a character but placed 
beside the associated arrow symbol, not above or below it. To which side? I 
haven't decided, but currently favour the left, at least in scores (as 
opposed to in text).

See Yahoo groups: /tuning-math/files/Dave/5Schismas.bmp *
for some examples.
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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Message: 5838

Date: Sun, 05 Jan 2003 06:03:22

Subject: Re: Poptimal generators

From: Dave Keenan

I think I actually followed most of that, and what's more, found it
very interesting. Thanks. 

But with regard to sagittal notation for these temperaments, some of
those results looked pretty wild, such as 205-ET for 7-limit Meantone.

What do you mean by rational generator of minimum height? What is
"height" here?


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Message: 5845

Date: Sun, 05 Jan 2003 10:17:03

Subject: Re: Poptimal generators

From: Carl Lumma

>"Height" is what number theorists call a function
>which measures the complexity of a rational (or
>algebraic) number.

What common feature would all such functions share?
IOW, how would they define "complexity"?

-C.


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Message: 5849

Date: Mon, 06 Jan 2003 13:40:01

Subject: Re: A common notation for JI and ETs

From: David C Keenan

In case anyone has already looked at the .bmp for my latest suggestion 
regarding symbolising the 5'-comma (5-schisma) up and down:

It was riddled with vertical alignment errors so I've had another go at it.

See Yahoo groups: /tuning-math/files/Dave/5Schismas.bmp *
-- Dave Keenan
Brisbane, Australia
Dave Keenan's Home Page *


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