Partch's scales on the Miracle keyboard

Yes folks, this is it! George Secor published THE general-purpose 11-limit
keyboard design in 1975 and it has languished in Xenharmonikon for 26 years!

I thought some folk might be better able to follow the correspondence
between MIRACLE and Partch's various scales if they are shown on Secor's
MIRACLE keyboard.

This draws on both
http://www.anaphoria.com/secor.PDF
and
http://x31eq.com/decimal_lattice.htm#partch

Unlike Secor, I'm showing exactly one octave. In real life, the rectangular
grid shown below would be rotated a few degrees clockwise so that the same
notes in different octaves are the same height (distance from the player)
and skewed so that pitch strictly increases from left to right. See the
black-and-white dots on
http://dkeenan.com/Music/BlackjackKeyboard.gif
or
http://dkeenan.com/Music/BlackjackKeyboard.doc

In decimal notation the Miracle keyboard looks like this (showing the full
MIRACLE-72 or 72EDO).
9>>>>
0>>> 1>>> 2>>> 3>>> 4>>> 5>>> 6>>> 7>>> 8>>> 9>>>
0>> 1>> 2>> 3>> 4>> 5>> 6>> 7>> 8>> 9>>
0> 1> 2> 3> 4> 5> 6> 7> 8> 9>
0 1 2 3 4 5 6 7 8 9
0< 1< 2< 3< 4< 5< 6< 7< 8< 9<
0<< 1<< 2<< 3<< 4<< 5<< 6<< 7<< 8<< 9<<
0<<< 1<<< 2<<< 3<<< 4<<< 5<<< 6<<< 7<<< 8<<< 9<<<
0<<<<

In 6*12 notation, with Partch's "G" for 1/1 (mapped to 0 decimal), it looks
like this

Legend:
A,B,C,D,E,F,G,b,# as in 12-tET
] = 1/4 tone up
> = 1/6 tone up
^ = 1/12 tone up
v = 1/12 tone down
< = 1/6 tone down
[ = 1/4 tone down
Gv
G# A^ Bb> B] C#< Dv Eb E^ F> G[
G#< Av Bb B^ C> D[ Eb< Ev F F#^
G> A[ Bb< Bv C C#^ D> E[ F< F#v
G G#^ A> B[ C< C#v D Eb^ E> F]
G< G#v A Bb^ B> C] D< Ebv E F^
F#> G] A< Bbv B C^ C#> D] E< Fv
F# G^ G#> A] B< Cv C# D^ Eb> E]
F#<

Horizontally we go in steps of 7/72 octave (MIRACLE generators) and
vertically we go in steps of 2/72 octave.

Here I've skewed it so you can see the pitch strictly increasing from left
to right, in proper keyboard fashion.

Gv
G# A^ Bb> B] C#< Dv Eb E^ F>
G[
G#< Av Bb B^ C> D[ Eb< Ev F
F#^
G> A[ Bb< Bv C C#^ D> E[ F< F#v
G G#^ A> B[ C< C#v D Eb^ E> F]
G< G#v A Bb^ B> C] D< Ebv E F^
F#> G] A< Bbv B C^ C#> D] E< Fv
F# G^ G#> A] B< Cv C# D^ Eb> E]
F#<

The above may have been scrambled by Yahoo-insterted line breaks (you can
try widening your window), so I'll stick to the rectangular array for the
rest of this post.

Now that you hopefully understand what the MIRACLE keyboard is about,
here's how the 11-limit tonality diamond (29 notes) maps to it.
20/11
10/9 14/11 16/11 14/9 5/3 16/9
12/11 7/6 5/4 4/3 10/7 18/11 7/4
1/1 8/7 11/9 7/5 3/2 8/5 12/7 11/6
9/8 6/5 9/7 11/8 11/7 9/5
11/10

It has only two notes which are outside the 41-note MIRACLE MOS (11/10 and
20/11). Harry Partch knew nothing of the Miracle temperament or the
corresponding keyboard when he proceded to fill in the melodic gaps
(although he was presumably looking for harmonically (JI) useful notes to
do it with).

Now that we know about Miracle, we could tell him that the most
harmonically productive way to even it out is probably by adding those
notes missing from the MIRACLE-41 MOS. In JI terms, this means mapping
notes to the appropriate gaps on the Miracle keyboard which are a 14:15 or
15:16 or 72:77 (or even a 176:189) away from their neighbours. We might
have ended up with 43 notes as follows.

20/11
28/27 10/9 32/27 14/11 15/11 16/11 14/9 5/3 16/9 40/21
64/63 12/11 7/6 5/4 4/3 10/7 32/21 18/11 7/4 15/8
1/1 16/15 8/7 11/9 21/16 7/5 3/2 8/5 12/7 11/6
63/32 21/20 9/8 6/5 9/7 11/8 22/15 11/7 27/16 9/5
27/14 11/10

Or we might suggest the following, which doesn't contain a continuous
Miracle-41 but is slightly more even.
20/11 64/33
10/9 32/27 14/11 15/11 16/11 14/9 5/3 16/9 40/21
64/63 12/11 7/6 5/4 4/3 10/7 32/21 18/11 7/4 15/8
1/1 16/15 8/7 11/9 21/16 7/5 3/2 8/5 12/7 11/6
63/32 21/20 9/8 6/5 9/7 11/8 22/15 11/7 27/16 9/5
33/32 11/10

Note that several different ratios could be chosen for any given Miracle
key. For example, 49/48 could replace 64/63, and 22/21 could replace 21/20,
as they do in some scales below.

In the Scala archive
http://www.xs4all.nl/~huygensf/doc/scales.zip
we find that Partch tried various ways of filling in the gaps, involving
37, 39, 41 and 43 notes.

Here they are on the Miracle keyboard.

! partch_37.scl
!
From "Exposition on Monophony" 1933, unp. see Ayers, 1/1 vol.9(2)

20/11 64/33
10/9 14/11 16/11 14/9 5/3 16/9 21/11
49/48 12/11 7/6 5/4 4/3 10/7 18/11 7/4 15/8
1/1 16/15 8/7 11/9 7/5 3/2 8/5 12/7 11/6
96/49 22/21 9/8 6/5 9/7 11/8 11/7 9/5
33/32 11/10

! partch_39.scl
!
Ur-Partch Keyboard 39 tones, published in Interval
64/33
10/9 14/11 15/11 16/11 14/9 5/3 16/9 21/11
49/48 12/11 7/6 5/4 4/3 10/7 32/21 18/11 7/4 15/8
1/1 16/15 8/7 11/9 21/16 7/5 3/2 8/5 12/7 11/6
96/49 22/21 9/8 6/5 9/7 11/8 22/15 11/7 9/5
33/32

Note that here he's dropped the 11/10 and 20/11 from the diamond; a very
"Miracle" thing to do. If he'd replaced 33/32 with 28/27 (and
correspondingly for its inversion) it would be a subset of Miracle-41.

! partch_41a.scl
!
From "Exposition on Monophony" 1933, unp. see Ayers, 1/1 vol. 9(2)

20/11 64/33
10/9 14/11 15/11 16/11 14/9 5/3 16/9 21/11
49/48 12/11 7/6 5/4 4/3 10/7 32/21 18/11 7/4 15/8
1/1 16/15 8/7 11/9 21/16 7/5 3/2 8/5 12/7 11/6
96/49 22/21 9/8 6/5 9/7 11/8 22/15 11/7 9/5
33/32 11/10

It's the same as the 39 tone scale above, but with 11/10 and 20/11 back again.

! PARTCH_43.scl
!
Harry Partch's 43-tone pure scale

160/81
40/27 20/11 64/33
10/9 32/27 14/11 16/11 14/9 5/3 16/9 40/21
12/11 7/6 5/4 4/3 10/7 32/21 18/11 7/4 15/8
1/1 16/15 8/7 11/9 21/16 7/5 3/2 8/5 12/7 11/6
21/20 9/8 6/5 9/7 11/8 11/7 27/16 9/5
33/32 11/10 27/20
81/80

It seems odd to have 81/80 instead of 49/48 or 64/63, and 27/20 instead of
15/11. Anyone know if this scale came before or after the one above? This
is the one Secor used in his article.

! partch_43a.scl
!
From "Exposition on Monophony" 1933, unp. see Ayers, 1/1 vol. 9(2)

20/11 64/33
10/9 32/27 14/11 15/11 16/11 14/9 5/3 16/9 40/21
49/48 12/11 7/6 5/4 4/3 10/7 32/21 18/11 7/4 15/8
1/1 16/15 8/7 11/9 21/16 7/5 3/2 8/5 12/7 11/6
96/49 21/20 9/8 6/5 9/7 11/8 22/15 11/7 27/16 9/5
33/32 11/10

Well whadya know? A perfectly "Miracle" way of fleshing out the 11-limit
diamond!

It turns out that Secor had the worst Partch scale to map to the Miracle
keyboard.

The main things to notice about all the above scales is that Partch never
wants two notes that would map to the same Miracle key, and his scales are
very compact on the Miracle keyboard. No overloads, very few holes.

Note that Miracle is able to distinguish 11/10 from both 10/9 and 12/11, as
Partch does. This contrasts with Erv Wilson's suggestion that Partch's
scale might be mapped to 41-EDO (or some such).

Anyone want to try mapping the rest of Partch's scales from the Scala
archive. It will be interesting to see if the 13-limit diamond has
overloads on the Miracle keyboard (I expect it does). But Partch's 11-limit
scales with fewer than 29 notes would be of interest too.

Then we could check out all Wilson's 11-limit scales. How come D'Alessandro
isn't in the Scala archive?

--------------------------------------------
How to place a ratio on the Miracle keyboard
--------------------------------------------

If you want to know where a ratio goes on the Miracle keyboard,
prime-factorise it (numerator and denominator) and then replace prime
factors with numbers of generators as follows

Prime No. gens
---------------
2 0
3 6
5 -7
7 -2
11 15
13 * -3 * poorly approximated by Miracle temperament
17 * 1 but still consistently mapped on the keyboard

Then add up the generators in the numerator and subtract those in the
denominator. e.g. 49 7 * 7 -2 + -2
-- = ----------------- -> ------------- -> -4 - 6 = -10
48 3 * 2 * 2 * 2 * 2 6 + 0 + ...

This gives you the number of generators that the key is from 1/1.

Some bright fellow could write a program that runs through the Scala
archive, and for 17-limit rational scales, finds all those that don't map
two different notes to the same Miracle note (i.e. no overloads), and tells
us the span of the Scale on the Miracle chain, with the scale's cardinality
for comparison. The we'd see JUST how useful this keyboard is, and how far
short of the full 72 it could usefully be made.

Can this Miracle layout be mapped efficiently to the Microzone?

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

--- In tuning@y..., David C Keenan <D.KEENAN@U...> wrote:
> Yes folks, this is it! George Secor published THE general-purpose
11-limit
> keyboard design in 1975 and it has languished in Xenharmonikon for
26 years!

You know, I'm always jealous of the few people I've met (like Robin
Pitre and Joseph Pehrson) who sprung for the full set of XHs. The big
academic libraries have got to wake up!

> This draws on both
> http://www.anaphoria.com/secor.PDF

So George proposed 116.69 cents as his version of the "MIRACLE"
generator. Is this the 11-limit minimax optimum?

> It has only two notes which are outside the 41-note MIRACLE MOS
(11/10 and
> 20/11).

Yes. These are members of the diamond so Partch was particularly
reluctant to give them up.

> Harry Partch knew nothing of the Miracle temperament or the
> corresponding keyboard when he proceded to fill in the melodic gaps
> (although he was presumably looking for harmonically (JI) useful
notes to
> do it with).

Yes.
>
> Now that we know about Miracle, we could tell him that the most
> harmonically productive way to even it out is probably by adding
those
> notes missing from the MIRACLE-41 MOS.

He may not agree with us on "harmonically productive" . . . the scale
he settled upon in _Genesis_ suggests somewhat differently.
>
> Note that here he's dropped the 11/10 and 20/11 from the diamond; a
very
> "Miracle" thing to do.

Nah . . . it's merely a very "41" thing to do -- and many, many
generators (such as the perfect fifth, for example) have MOSs at 41.
41-tET conflates 11/10 with 10/9, and 20/11 with 9/5, but otherwise
represents the harmonies quite well.

> If he'd replaced 33/32 with 28/27 (and
> correspondingly for its inversion) it would be a subset of Miracle-
> 41.

To within a few cents, correct?
>
> ! PARTCH_43.scl
> !
> Harry Partch's 43-tone pure scale
>
>
160/81
> 40/27 20/11
64/33
> 10/9 32/27 14/11 16/11 14/9 5/3 16/9
40/21
> 12/11 7/6 5/4 4/3 10/7 32/21 18/11 7/4 15/8
> 1/1 16/15 8/7 11/9 21/16 7/5 3/2 8/5 12/7 11/6
> 21/20 9/8 6/5 9/7 11/8 11/7 27/16 9/5
> 33/32 11/10 27/20
> 81/80
>
> It seems odd to have 81/80 instead of 49/48 or 64/63, and 27/20
instead of
> 15/11. Anyone know if this scale came before or after the one
above? This
> is the one Secor used in his article.

This is the latest, "official" Partch scale, which he settled on for
the writing of his book, _Genesis of a Music_. You can see it
compared with a Fokker 41-tone periodicity block near the bottom of

http://www.ixpres.com/interval/td/erlich/partchpblock.htm

>
> ! partch_43a.scl
> !
> From "Exposition on Monophony" 1933, unp. see Ayers, 1/1 vol. 9(2)
>
>
> 20/11
64/33
> 10/9 32/27 14/11 15/11 16/11 14/9 5/3 16/9
40/21
> 49/48 12/11 7/6 5/4 4/3 10/7 32/21 18/11 7/4 15/8
> 1/1 16/15 8/7 11/9 21/16 7/5 3/2 8/5 12/7 11/6
> 96/49 21/20 9/8 6/5 9/7 11/8 22/15 11/7 27/16 9/5
> 33/32 11/10
>
> Well whadya know? A perfectly "Miracle" way of fleshing out the 11-
limit
> diamond!

That's interesting . . .
>
> It turns out that Secor had the worst Partch scale to map to the
Miracle
> keyboard.

Well, you might be able to argue that Partch was indirectly guided by
MIRACLE forces in 1933, but these fell by the wayside later.
>
> The main things to notice about all the above scales is that Partch
never
> wants two notes that would map to the same Miracle key,

Well, that would mean two notes very close in pitch!

> and his scales are
> very compact on the Miracle keyboard. No overloads, very few holes.

I think the reason for this is the fact that MIRACLE is so special
for generating 11-limit harmony (no generators give the hexad with
fewer notes with as much accuracy, or with as many notes with greater
accuracy, right?), and Partch's scale is rich in 11-limit harmony.
I'd be hesitant in drawing any further implications along these lines.

> Note that Miracle is able to distinguish 11/10 from both 10/9 and
12/11, as
> Partch does. This contrasts with Erv Wilson's suggestion that
Partch's
> scale might be mapped to 41-EDO (or some such).

Were they not both in the diamond, evidence seems to suggest that
Partch would have immediately dropped one of them (he even did so at
one point).
>
> Anyone want to try mapping the rest of Partch's scales from the
Scala
> archive.

I think you should write a paper for Xenharmonikon!

> It will be interesting to see if the 13-limit diamond has
> overloads on the Miracle keyboard (I expect it does).

Obviously, since 72-tET is not unique through the 13-limit but only
through the 11-limit.

Paul!
Thought you would want to at least see this Starting on 6 with the meat on 7 and 8.
http://www.anaphoria.com/tres.PDF

Paul Erlich wrote:

> > It will be interesting to see if the 13-limit diamond has
> > overloads on the Miracle keyboard (I expect it does).
>
> Obviously, since 72-tET is not unique through the 13-limit but only
> through the 11-limit.

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> Paul!
> Thought you would want to at least see this Starting on 6 with
the meat on 7 and 8.
> http://www.anaphoria.com/tres.PDF

Thanks Kraig! Dave, do these change anything?

Monz -- you should modify your MIRACLE webpages to state that George
Secor originally proposed the 116.69 cent generator in XH3. George
did not, however, mention any of the MOSs of this generator, focusing
instead on a slightly discontiguous set which very closely
approximated Partch's scale. Secor did, however, notice that a
keyboard built around this generator would function consistently in
31-tET, 41-tET, and 72-tET, so identifying the MIRACLE MOSs of those
cardinalities was just around the corner.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> So George proposed 116.69 cents as his version of the "MIRACLE"
> generator. Is this the 11-limit minimax optimum?

Oddly enough, no. He gives the correct minimax error of 3.32 cents,
but that actually occurs with a generator of 116.716 cents. Maybe he
gave the wrong generator by mistake. The RMS optimum is 116.678 cents
giving a maximum error of 3.77 cents.

> > If he'd replaced 33/32 with 28/27 (and
> > correspondingly for its inversion) it would be a subset of
Miracle-
> > 41.
>
> To within a few cents, correct?

Yes, on the 11-limit consonant intervals, but not the pitches.

> This is the latest, "official" Partch scale, which he settled on for
> the writing of his book, _Genesis of a Music_. You can see it
> compared with a Fokker 41-tone periodicity block near the bottom of
>
> http://www.ixpres.com/interval/td/erlich/partchpblock.htm

Thanks for that. How does that PB compare with Miracle-41?

> Well, you might be able to argue that Partch was indirectly guided
by
> MIRACLE forces in 1933, but these fell by the wayside later.

Yes. It seems he came to want more or longer chains of fifths.

> > The main things to notice about all the above scales is that
Partch
> never
> > wants two notes that would map to the same Miracle key,
>
> Well, that would mean two notes very close in pitch!

Hmm. How close? What's the _biggest_ 11-limit unison vector in
Miracle?

> > and his scales are
> > very compact on the Miracle keyboard. No overloads, very few
holes.
>
> I think the reason for this is the fact that MIRACLE is so special
> for generating 11-limit harmony (no generators give the hexad with
> fewer notes with as much accuracy, or with as many notes with
greater
> accuracy, right?)

Right.

>, and Partch's scale is rich in 11-limit harmony.
> I'd be hesitant in drawing any further implications along these
lines.
>

Fair enough. But to be unconsciously drawn to Miracle (at least in his
early years) it suggests he was making use of many harmonies that were
only approximately JI. So much for his "no temperaments" stance.

I remember being puzzled by this attitude in Carl Lumma at one time
too. To charicature it: "It's ok to use chords that contain errors of
an entire 224:225 or 384:385, but to distribute the errors so they are
less than 3.4 cents, oh no, that would be tempering".

> > Note that Miracle is able to distinguish 11/10 from both 10/9 and
> 12/11, as
> > Partch does. This contrasts with Erv Wilson's suggestion that
> Partch's
> > scale might be mapped to 41-EDO (or some such).
>
> Were they not both in the diamond, evidence seems to suggest that
> Partch would have immediately dropped one of them (he even did so at
> one point).
> >
> > Anyone want to try mapping the rest of Partch's scales from the
> Scala
> > archive.
>
> I think you should write a paper for Xenharmonikon!

You mean one based on the previous post, showing the Miracle keyboard
mappings, or what?
-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > > The main things to notice about all the above scales is that
> Partch
> > never
> > > wants two notes that would map to the same Miracle key,
> >
> > Well, that would mean two notes very close in pitch!
>
> Hmm. How close? What's the _biggest_ 11-limit unison vector in
> Miracle?

Just answering my own question here. Here's a big one:
2816:2835 = (3^4 * 5 * 7)/(2^8 * 11) ~= 11.6 cents. Partch had some
steps in his scales close to that (about 14 cents I think). So I still
think "no miracle overloads" may be significant. Maybe proponents of
alternative temperaments for Partch (schismic, kleismic), should check
for overloads.
-- Dave Keenan

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

> Thanks for that. How does that PB compare with Miracle-41?

What sort of comparison would you like to see? I wish we could
visualize them directly in 4D . . .
> > >
> > Well, that would mean two notes very close in pitch!
>
> Hmm. How close? What's the _biggest_ 11-limit unison vector in
> Miracle?

Arbitrarily large, if you combine lots of 385:384s and 225:224s the
right way. But within MIRACLE-72, probably not very large?
>
> > > and his scales are
> > > very compact on the Miracle keyboard. No overloads, very few
> holes.
> >
> > I think the reason for this is the fact that MIRACLE is so
special
> > for generating 11-limit harmony (no generators give the hexad
with
> > fewer notes with as much accuracy, or with as many notes with
> greater
> > accuracy, right?)
>
> Right.
>
> >, and Partch's scale is rich in 11-limit harmony.
> > I'd be hesitant in drawing any further implications along these
> lines.
> >
>
> Fair enough. But to be unconsciously drawn to Miracle (at least in
his
> early years) it suggests he was making use of many harmonies that
were
> only approximately JI.

We've heard of this before, through Wilson.
>
> I remember being puzzled by this attitude in Carl Lumma at one time
> too. To charicature it: "It's ok to use chords that contain errors
of
> an entire 224:225 or 384:385, but to distribute the errors so they
are
> less than 3.4 cents, oh no, that would be tempering".

Well, I guess some people prefer to have most of the chords just and
then have a few wolves, rather than having all the chords near-just.
If total pain is the sum of the pains of each chord, and the pain of
each chord is the sum of the pains of each interval, and the pain of
each interval increases as the square of the error, then Lumma's
attitude is wrong. But if the pain of each interval increases as,
say, the logarithm of the error, then Lumma's attitude is right.
Oops, that should've been on tuning-math@yahoogroups.com (you can
send further discussions there).

> > > Anyone want to try mapping the rest of Partch's scales from the
> > Scala
> > > archive.
> >
> > I think you should write a paper for Xenharmonikon!
>
> You mean one based on the previous post, showing the Miracle
keyboard
> mappings, or what?

Yes, and about MIRACLE in general. Graham and I would be co-authors.
(I don't think many other journals would be interested in an article
mainly about MIRACLE, but it seems that that's what you want to
write). John Chalmers told be he'll try to get XH 18 out by the end
of the year. I should write two papers for it, one on harmonic
entropy, one on periodicity blocks.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> > Paul!
> > Thought you would want to at least see this Starting on 6 with
> the meat on 7 and 8.
> > http://www.anaphoria.com/tres.PDF
>
> Thanks Kraig! Dave, do these change anything?

How should they? On p6, I see 11/10 treated as a second-class citizen,
this doesn't have to happen with a miracle keyboard mapping. What is
the difference between p6 and p7? I don't see the relevance of p8.
-- Dave Keenan

Dave,

We need a clarification here...

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> But to be unconsciously drawn to Miracle (at least in his
> early years) it suggests he was making use of many harmonies that
> were only approximately JI. So much for his "no temperaments"
> stance.

How you can take a construct that exists only recently, or exists
only as far back as the XH article, and 'map' it backwards in time to
account for Partch being "unconsciously drawn" to it? The Miracle
tuning paradigm certainly works for you and many of our other
members, and seems to hold bright promise, but why is it not simply
parallel developments with striking similarities?

As for making use of "many harmonies", could you point out examples
in his work, both early and mature, where you find this in evidence?

For Partch to have spent so much time researching, developing, and
then *using* his particular flavor of JI and then years later to
assume he should have or could have known and used a subset of an ET
is, well, a bit odd. There are certainly big improvements in all
facets of tuning knowledge since then, and some great strides are
being made. I don't know why this has to change history, however.

Besides, he did all the heavy lifting in the early days, and pretty
much from the mid-40's on he worked on composing and performing,
*not* theorizing. Sort of a "find it, tweak it, use it" paradigm.

Works for me. And if you get that guitar built, you too.

One last point: the thread is Partch's scales on the Miracle
keyboard, and yet his earliest work, especially the work in
development, started without any keyboard at all. I'm happy to be
corrected, but the first truly functional reed organ he retuned was
in the early 1940's (his first attempt in California unsuccesful),
and already he was composing and working in the 11-limit JI 'scale'
(as you put it) system. Mapping it now to a new keyboard is a great
idea, but certainly his own groundwork did *not* spring from a
keyboard oriented approach. (I may be reading too much into the
thread title, but just being clear in my intentions...)

Cheers,
Jon

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> > Thanks for that. How does that PB compare with Miracle-41?
>
> What sort of comparison would you like to see? I wish we could
> visualize them directly in 4D . . .

I guess what I want to know is, can Miracle-41 be considered a
tempering of that PB, or is there no relationship between their unison
vectors.

> > > >
> > > Well, that would mean two notes very close in pitch!
> >
> > Hmm. How close? What's the _biggest_ 11-limit unison vector in
> > Miracle?
>
> Arbitrarily large, if you combine lots of 385:384s and 225:224s the
> right way. But within MIRACLE-72, probably not very large?
> >
> > > > and his scales are
> > > > very compact on the Miracle keyboard. No overloads, very few
> > holes.
> > >
> > > I think the reason for this is the fact that MIRACLE is so
> special
> > > for generating 11-limit harmony (no generators give the hexad
> with
> > > fewer notes with as much accuracy, or with as many notes with
> > greater
> > > accuracy, right?)
> >
> > Right.
> >
> > >, and Partch's scale is rich in 11-limit harmony.
> > > I'd be hesitant in drawing any further implications along these
> > lines.
> > >
> >
> > Fair enough. But to be unconsciously drawn to Miracle (at least in
> his
> > early years) it suggests he was making use of many harmonies that
> were
> > only approximately JI.
>
> We've heard of this before, through Wilson.

Aha!

> > I remember being puzzled by this attitude in Carl Lumma at one
time
> > too. To charicature it: "It's ok to use chords that contain errors
> of
> > an entire 224:225 or 384:385, but to distribute the errors so they
> are
> > less than 3.4 cents, oh no, that would be tempering".

> Well, I guess some people prefer to have most of the chords just and
> then have a few wolves, rather than having all the chords near-just.

Yes. But when the errors we're talking about are the same as typical
acoustic instrument intonation errors? And there's a whole world of
"well" temepraments and planar temperaments before you go to linear
ones.

> > > I think you should write a paper for Xenharmonikon!
> >
> > You mean one based on the previous post, showing the Miracle
> keyboard
> > mappings, or what?
>
> Yes, and about MIRACLE in general. Graham and I would be co-authors.
> (I don't think many other journals would be interested in an article
> mainly about MIRACLE, but it seems that that's what you want to
> write).

Yes.

Dave and Paul!
I feel you are playing some mental games here. You can't say Partch wanted 11 limit scales and
then come up with a very close appox imitation and say he was drawn to that. It seems Partch
created the diamond first and then filled out the gaps in his tuning that would make it a Constant
Structure, 41 tones with 2 variables. As the Lambdoma is an Ancient structure going back to at
least the Greeks, It was apart of his artistic statement to draw upon this tradition to redirect
music's course as the the first historical point in the past to safely use as a point of
departure.

Dave Keenan wrote:

> Fair enough. But to be unconsciously drawn to Miracle (at least in his
> early years) it suggests he was making use of many harmonies that were
> only approximately JI. So much for his "no temperaments" stance.

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., "Jon Szanto" <JSZANTO@A...> wrote:
> Dave,
>
> We need a clarification here...
>
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > But to be unconsciously drawn to Miracle (at least in his
> > early years) it suggests he was making use of many harmonies that
> > were only approximately JI. So much for his "no temperaments"
> > stance.
>
> How you can take a construct that exists only recently, or exists
> only as far back as the XH article, and 'map' it backwards in time
to
> account for Partch being "unconsciously drawn" to it? The Miracle
> tuning paradigm certainly works for you and many of our other
> members, and seems to hold bright promise, but why is it not simply
> parallel developments with striking similarities?

That's the point of view I've been trying to counter Dave with, Jon.
But . . .
>
> As for making use of "many harmonies", could you point out examples
> in his work, both early and mature, where you find this in evidence?

We've heard evidence of this via Erv Wilson. Partch was asking Wilson
about certain harmonies on the Chromelodeon that he didn't
understand, and Wilson explained, showing that Partch was in fact
using the near-just sonorities (of which Partch had done relatively
little study) in his scale as if they were just. Hence, Wilson's
keyboard designs allow for these sonorities to be played with the
same fingerings as the just ones.
>
> For Partch to have spent so much time researching, developing, and
> then *using* his particular flavor of JI and then years later to
> assume he should have or could have known and used a subset of an
ET
> is, well, a bit odd.

No subsets of ETs are being explicitly put forth here, though we have
anecdotal evidence (again via Wilson) that Partch accepted 41-tET
(presumably in some particular context) as aurally identical to his
scale, and we know that 72-tET is even more accurate.

> There are certainly big improvements in all
> facets of tuning knowledge since then, and some great strides are
> being made. I don't know why this has to change history, however.

What do you mean by "change history"? Certainly I agree that Dave may
be overestimating the importance of MIRACLE in Partch's thought.
We're having a concurrent discussion on this on

tuning-math@yahoogroups.com

.

> One last point: the thread is Partch's scales on the Miracle
> keyboard, and yet his earliest work, especially the work in
> development, started without any keyboard at all. I'm happy to be
> corrected, but the first truly functional reed organ he retuned was
> in the early 1940's (his first attempt in California unsuccesful),
> and already he was composing and working in the 11-limit JI 'scale'
> (as you put it) system. Mapping it now to a new keyboard is a great
> idea, but certainly his own groundwork did *not* spring from a
> keyboard oriented approach. (I may be reading too much into the
> thread title

I'm afraid it sure looks like you are. Clearly, others before us have
been interested in having an easily usable and understandable layout
of Partch's scale on a keyboard, thus the excellent work that Secor
and Wilson did in this area. But keyboard or no keyboard, there are a
lot of quasi-just sonorities in Partch's scale that he seems to have
been semi-conscious of, and that one may want to use, whether one
tempers them out or leaves them as "microwolves".

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> >
> > > Thanks for that. How does that PB compare with Miracle-41?
> >
> > What sort of comparison would you like to see? I wish we could
> > visualize them directly in 4D . . .
>
> I guess what I want to know is, can Miracle-41 be considered a
> tempering of that PB, or is there no relationship between their
unison
> vectors.

Is there a comprehensive way to find an answer to this question? I
can spit out the FPB for any set of unison vectors in no time . . .
please instruct (brain sugar low right now).

> > > Fair enough. But to be unconsciously drawn to Miracle (at least
in
> > his
> > > early years) it suggests he was making use of many harmonies
that
> > were
> > > only approximately JI.
> >
> > We've heard of this before, through Wilson.
>
> Aha!

Yes, but as far as I know, Wilson only went as far as to recognize
these as modulus-41 equivalencies. Whether they were all MIRACLE
equivalencies too, I don't know, and we may never know for sure . . .

> Yes. But when the errors we're talking about are the same as
typical
> acoustic instrument intonation errors?

Very good point, given that Partch used acoustic instruments (mostly
ones without harmonic partials!!)

> And there's a whole world of
> "well" temepraments and planar temperaments before you go to linear
> ones.

You bet.

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> Dave and Paul!
> I feel you are playing some mental games here. You can't say
Partch wanted 11 limit scales and
> then come up with a very close appox imitation and say he was drawn
to that.

Exactly my response in _opposition_ to Dave.

> It seems Partch
> created the diamond first and then filled out the gaps in his
tuning that would make it a Constant
> Structure, 41 tones with 2 variables.

That's exactly the tack I've been taking, in _opposition_ to Dave!
You haven't been reading very closely, I fear. Some of the exchanges
have been on the tuning-math list, so you can't be blamed for that.

> As the Lambdoma is an Ancient structure going back to at
> least the Greeks, It was apart of his artistic statement to draw
upon this tradition to redirect
> music's course as the the first historical point in the past to
safely use as a point of
> departure.

The importance of the Diamond -- just the point I keep making in
_opposition_ to Dave.

--- In tuning@y..., "Jon Szanto" <JSZANTO@A...> wrote:
> Dave,
>
> We need a clarification here...
>
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > But to be unconsciously drawn to Miracle (at least in his
> > early years) it suggests he was making use of many harmonies that
> > were only approximately JI. So much for his "no temperaments"
> > stance.
>
> How you can take a construct that exists only recently, or exists
> only as far back as the XH article, and 'map' it backwards in time
to
> account for Partch being "unconsciously drawn" to it? The Miracle
> tuning paradigm certainly works for you and many of our other
> members, and seems to hold bright promise, but why is it not simply
> parallel developments with striking similarities?

In a sense Miracle has been there forever, what happened recently is
that we became conscious of it. If Partch was trying out candidates to
fill in the melodic holes in the 11-limit diamond, _and_ he was
willing to use some approx JI harmonies with 224:225 and 384:385
errors (less than 8 cents) then a search by ear for candidates with
maximum 11-limit relationships to the notes already present, would
automatically tend to fill out a (rationalised) Miracle chain.

> As for making use of "many harmonies", could you point out examples
> in his work, both early and mature, where you find this in evidence?

I didn't _say_ he did. I asked others who have knowledge and access to
these, to _tell_ me if he did. I think Paul Erlich just told us (I
forget whether it was in this list or tuning-math) that Erv Wilson
said Partch did use these.

> For Partch to have spent so much time researching, developing, and
> then *using* his particular flavor of JI and then years later to
> assume he should have or could have known and used a subset of an ET
> is, well, a bit odd.

I don't think I wrote any such thing, except "could have used".

> There are certainly big improvements in all
> facets of tuning knowledge since then, and some great strides are
> being made. I don't know why this has to change history, however.

I think you're seriously misreading me. I intended no such
thing. Perhaps if you point me to the offending passage(s) in what I
wrote, I can straighten it out.

> Mapping it now to a new keyboard is a great
> idea, but certainly his own groundwork did *not* spring from a
> keyboard oriented approach. (I may be reading too much into the
> thread title, but just being clear in my intentions...)

No. I'm not suggesting the keyboard suggested Miracle fill ins. I'm
suggesting the _ear_ did.
-- Dave Keenan

Paul Erlich wrote:

> We've heard evidence of this via Erv Wilson. Partch was asking Wilson
> about certain harmonies on the Chromelodeon that he didn't
> understand, and Wilson explained, showing that Partch was in fact
> using the near-just sonorities (of which Partch had done relatively
> little study) in his scale as if they were just. Hence, Wilson's
> keyboard designs allow for these sonorities to be played with the
> same fingerings as the just ones.

I believe it was on one of the marimbas and the context was melodic

> No subsets of ETs are being explicitly put forth here, though we have
> anecdotal evidence (again via Wilson) that Partch accepted 41-tET
> (presumably in some particular context) as aurally identical to his
> scale, and we know that 72-tET is even more accurate.

Erv only showed that Partch was hearing a 41 tone scale. A constant structure with two variables!

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> Kraig Grady wrote:
> > As the Lambdoma is an Ancient structure going back to at
> > least the Greeks, It was apart of his artistic statement to draw
> upon this tradition to redirect
> > music's course as the the first historical point in the past to
> safely use as a point of
> > departure.
>
> The importance of the Diamond -- just the point I keep making in
> _opposition_ to Dave.

Huh? I haven't denied the importance of the diamond. I see that it's
the "41-tones with 2 variables" camp that is denying the importance of
the diamond, by treating 11/10 and 20/11 as second-class citizens.
-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> I didn't _say_ he did. I asked others who have knowledge and access
to
> these, to _tell_ me if he did. I think Paul Erlich just told us (I
> forget whether it was in this list or tuning-math) that Erv Wilson
> said Partch did use these.

I think these anecdotes originally came to us from Wilson through
Daniel Wolf (who has been having e-mail problems since well before
the Zill fiasco). Also, Daniel Wolf has done some in-depth analyses
of Partch scores, so his opinion here would carry a lot of weight.

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

> Huh? I haven't denied the importance of the diamond. I see that
it's
> the "41-tones with 2 variables" camp that is denying the importance
of
> the diamond, by treating 11/10 and 20/11 as second-class citizens.

On the contrary, it's only by virtue of belonging to the diamond that
11/10 and 20/11 make it in to the scale at all. Otherwise we'd simply
have a 41-tone CS/PB. The only force strong enough to overcome
the "closure" attained at 41 is a diamondic force.

Dave!
Please define what a second-class citizen is. You have a harmonic construct (the diamond) and
a melodic Construct (41) and a kink in their resolution. Its just a reality

Dave Keenan wrote:

> Huh? I haven't denied the importance of the diamond. I see that it's
> the "41-tones with 2 variables" camp that is denying the importance of
> the diamond, by treating 11/10 and 20/11 as second-class citizens.

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

Paul,

Thanks for jumping in!

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Jon Szanto" <JSZANTO@A...> wrote:
> We've heard evidence of this via Erv Wilson. Partch was asking
> Wilson about certain harmonies on the Chromelodeon that he didn't
> understand, and Wilson explained, showing that Partch was in fact
> using the near-just sonorities (of which Partch had done relatively
> little study) in his scale as if they were just.

Ah, I see. OTOH, this couldn't have been any earlier than the mid-
60's or so, meaning the last decade of his life, with only a couple
of pieces (albeit significant ones) left to be composed. And already
dominated by percussive instruments, lending an emphasis on 'harmony'
somewhat off to the side.

(Aside: I always forget that people refer to Partch's "scale", which
would be the 43-tones all strung out in an octave. I've always
heard/seen that as the by-product of the Tonality Diamond laid out
sequentially, and not as the genesis (if you will) of the composing)

> Hence, Wilson's keyboard designs allow for these sonorities to be
> played with the same fingerings as the just ones.

Rats, too late for HP. But he also, I think, was using these 'nether
regions' of the JI construct, the bits that fell into place to
complete the diamond, as the spice in the JI tonality fields. The
extra colors on the pallette, so to speak. I am sure this is all
subject to more discussion...

> No subsets of ETs are being explicitly put forth here, though we
> have anecdotal evidence (again via Wilson) that Partch accepted
> 41-tET (presumably in some particular context) as aurally identical
> to his scale, and we know that 72-tET is even more accurate.

Right, I understand the latter bits. I thought the mappings of the M
were using subsets of the 72, but I guess I haven't followed closely.
As for the anecdotal bits, sure, but it is all pretty much after the
fact. Had Erv played 41-tET for Harry in 1930, it would mean a lot.
But whether you like it or not, some people (and I'd wager that HP
was one) would like JI for other aspects then just the raw 'aural
identities'. Call it a philosophical stance.

> What do you mean by "change history"? Certainly I agree that Dave
> may be overestimating the importance of MIRACLE in Partch's
> thought.

My point exactly, bucko! "MIRACLE in Partch's thought"? He wasn't
thinking of it, unless you want to rewrite, or go back and find
someone that said "You know, I showed HP this M scale and he
went 'Holy mackeral, that's boffo!'".

That's what I mean by change history, saying that Partch was
unconsciously drawn to something that hadn't been written on or
discussed.

> We're having a concurrent discussion on this on tuning-math@y...

No time.

> > (I may be reading too much into the thread title)

> I'm afraid it sure looks like you are. Clearly, others before us
> have been interested in having an easily usable and understandable
> layout of Partch's scale on a keyboard, thus the excellent work
> that Secor and Wilson did in this area. But keyboard or no
> keyboard, there are a lot of quasi-just sonorities in Partch's
> scale that he seems to have been semi-conscious of,

Hey, we've all been semi-conscious from time to time...

> and that one may want to use, whether one tempers them out or
> leaves them as "microwolves".

Or uses them as the wild things that they are.

Good illuminations, thanks,
Jon

Dave,

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> In a sense Miracle has been there forever, what happened recently
> is that we became conscious of it.

...or rather: Dave, Dave, Dave! That is just a bit cosmic for someone
as pragmatic as Partch. *Everything* has been there forever, if you
want to follow that line of reasoning. I'm just not buying the
concept that now that people have codified (or are in the process of)
the M stuff, that you can say past *musical* choices were guided by
this... this... 'unknown-but-existing construct'.

> If Partch was trying out candidates to
> fill in the melodic holes in the 11-limit diamond, _and_ he was
> willing to use some approx JI harmonies with 224:225 and 384:385
> errors

I don't understand why he would even be thinking to use some "approx
JI harmonies" when he could use them derived directly from the JI
construct.

> I didn't _say_ he did. I asked others who have knowledge and access
> to these, to _tell_ me if he did. I think Paul Erlich just told us
> (I forget whether it was in this list or tuning-math) that Erv
> Wilson said Partch did use these.

Replied to in Paul's ealier post...

> I don't think I wrote any such thing, except "could have used".

Sorry. As I explained to PE, I was mistaken in thinking that this was
a subset of M that was being discussed. Jet lag.

> I think you're seriously misreading me.

No, you'll know when I'm serious! <g>

> I intended no such thing. Perhaps if you point me to the offending
> passage(s) in what I wrote, I can straighten it out.

I think I explained pretty well in that previous post to PE about the
historical backward glances being committed herein.

> No. I'm not suggesting the keyboard suggested Miracle fill ins. I'm
> suggesting the _ear_ did.

Got it. Yep, he was ear-oriented, that is for sure.

Cheers,
Jon

D,

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> I see that it's the "41-tones with 2 variables" camp that is
> denying the importance of the diamond....

Easy on the "camp" metaphors, Dave. We seen enough fractional vitriol
for a few weeks, at least.

Cheers,
Jon

PE,

(I'll stop posting in just a second, I swear!)

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> The only force strong enough to overcome
> the "closure" attained at 41 is a diamondic force.

Do you realize how much that sounds like "demonic force"? :)

J

--- In tuning@y..., "Jon Szanto" <JSZANTO@A...> wrote:
>
> Rats, too late for HP. But he also, I think, was using
these 'nether
> regions' of the JI construct, the bits that fell into place to
> complete the diamond, as the spice in the JI tonality fields. The
> extra colors on the pallette, so to speak. I am sure this is all
> subject to more discussion...

The "spice" is what allowed Partch to modulate to hexads which did
not include 1/1 as a member.
>
> > No subsets of ETs are being explicitly put forth here, though we
> > have anecdotal evidence (again via Wilson) that Partch accepted
> > 41-tET (presumably in some particular context) as aurally
identical
> > to his scale, and we know that 72-tET is even more accurate.
>
> Right, I understand the latter bits. I thought the mappings of the
M
> were using subsets of the 72, but I guess I haven't followed
closely.

The MIRACLE generator describes various generators very close, but
non necessarily identical, to 7/72 octave.

> As for the anecdotal bits, sure, but it is all pretty much after
the
> fact. Had Erv played 41-tET for Harry in 1930, it would mean a lot.
> But whether you like it or not, some people (and I'd wager that HP
> was one) would like JI for other aspects then just the raw 'aural
> identities'. Call it a philosophical stance.

Ah . . . I've heard Dante Rosati and one other person approach JI in
this manner. And I respect it. Personally, for my own music, I
respect the common ear most highly of all, far more highly than any
theoretical abstraction.

> > and that one may want to use, whether one tempers them out or
> > leaves them as "microwolves".
>
> Or uses them as the wild things that they are.

Wouldn't that be the same as leaving them in as "microwolves"?

--- In tuning@y..., "Jon Szanto" <JSZANTO@A...> wrote:
> Dave,
>
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > In a sense Miracle has been there forever, what happened recently
> > is that we became conscious of it.
>
> ...or rather: Dave, Dave, Dave! That is just a bit cosmic for
someone
> as pragmatic as Partch. *Everything* has been there forever, if you
> want to follow that line of reasoning. I'm just not buying the
> concept that now that people have codified (or are in the process
of)
> the M stuff, that you can say past *musical* choices were guided by
> this... this... 'unknown-but-existing construct'.

Why not? I think this sort of thing happened many, many times in the
history of music and in every other field of musical endeavor. Though
I might disagree with Dave on some aspects of what he's claiming
here. What about Kraig's/Wilson's analysis that Partch created a 41-
tone CS with 2 variable tones? Certainly Partch wasn't consciously
aware of that . . . but the forces underlying the concept seem to
have guided him nonetheless.
>
> > If Partch was trying out candidates to
> > fill in the melodic holes in the 11-limit diamond, _and_ he was
> > willing to use some approx JI harmonies with 224:225 and 384:385
> > errors
>
> I don't understand why he would even be thinking to use
some "approx
> JI harmonies" when he could use them derived directly from the JI
> construct.

Because he couldn't! That's just the point! They aren't derived
directly from the JI construct, and there's no way to see them in the
ratios themselves, unless you specifically set out to multiply and
divide all of them by 224:225, 384:385, 539:540, etc., and see what
results. As far as we know, Partch didn't _consciously_ do this --
but he very well may have been guided by such considerations when
_aurally_ playing around with the various scales he was considering.

Did you read George Secor's article? Secor showed that in Partch's 43-
tone scale, there are actually 21 otonal hexads and 21 utonal hexads
if you allow these small errors (someone correct me if I'm
misunderstanding Secor). Even more hexads would be present in earlier
versions of Partch's scale. But Partch doesn't explicitly mention
the "extra" hexads at all in his writings, because in terms of strict-
JI math, they don't even exist.

PE!

Paul Erlich wrote:

What about Kraig's/Wilson's analysis that Partch created a 41-

> tone CS with 2 variable tones? Certainly Partch wasn't consciously
> aware of that . . . but the forces underlying the concept seem to
> have guided him nonetheless.

He was aware he was "filling in the gaps" in his scale. Also the concept of Moments of Symmetry
and Constant Structure are general principles. Outside of being a straight linear temperment
series I see no general principle in Miracle.

>
> >
> > > If Partch was trying out candidates to
> > > fill in the melodic holes in the 11-limit diamond, _and_ he was
> > > willing to use some approx JI harmonies with 224:225 and 384:385
> > > errors
> >
> > I don't understand why he would even be thinking to use
> some "approx
> > JI harmonies" when he could use them derived directly from the JI
> > construct.
>
> Because he couldn't! That's just the point! They aren't derived
> directly from the JI construct, and there's no way to see them in the
> ratios themselves, unless you specifically set out to multiply and
> divide all of them by 224:225, 384:385, 539:540, etc., and see what
> results. As far as we know, Partch didn't _consciously_ do this --
> but he very well may have been guided by such considerations when
> _aurally_ playing around with the various scales he was considering.
>
> Did you read George Secor's article? Secor showed that in Partch's 43-
> tone scale, there are actually 21 otonal hexads and 21 utonal hexads
> if you allow these small errors (someone correct me if I'm
> misunderstanding Secor). Even more hexads would be present in earlier
> versions of Partch's scale. But Partch doesn't explicitly mention
> the "extra" hexads at all in his writings, because in terms of strict-
> JI math, they don't even exist.

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> PE!
>
> Paul Erlich wrote:
>
> What about Kraig's/Wilson's analysis that Partch created a 41-
>
> > tone CS with 2 variable tones? Certainly Partch wasn't consciously
> > aware of that . . . but the forces underlying the concept seem to
> > have guided him nonetheless.
>
> He was aware he was "filling in the gaps" in his scale. Also the
concept of Moments of Symmetry
> and Constant Structure are general principles. Outside of being a
straight linear temperment
> series I see no general principle in Miracle.

Which is why I thought Dave went too far. But I strongly disagree
with Jon that undiscovered principles can't exert any influence
before they've been discovered. The general principles you mention,
for example, were "at work" in shaping Partch's choices even though
he was not consciously aware of them in their full generality.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > I guess what I want to know is, can Miracle-41 be considered a
> > tempering of that PB, or is there no relationship between their
> unison
> > vectors.
>
> Is there a comprehensive way to find an answer to this question? I
> can spit out the FPB for any set of unison vectors in no time . . .
> please instruct (brain sugar low right now).

I dunno. Is it enough to find a set of 11-limit unison vectors for
Miracle-41 and see if they are linearly independent of those for
your 41 tone PB. I notice you use FPB for "Fokker periodicity block".
Is there any other kind?

> Yes, but as far as I know, Wilson only went as far as to recognize
> these as modulus-41 equivalencies. Whether they were all MIRACLE
> equivalencies too, I don't know, and we may never know for sure . .

Yeah, we're only speculating. Not claiming any "science" here. Not
trying to rewrite history. Ok everyone?

-- Dave Keenan

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> > Huh? I haven't denied the importance of the diamond. I see that
> it's
> > the "41-tones with 2 variables" camp that is denying the
importance
> of
> > the diamond, by treating 11/10 and 20/11 as second-class citizens.
>
> On the contrary, it's only by virtue of belonging to the diamond
that
> 11/10 and 20/11 make it in to the scale at all. Otherwise we'd
simply
> have a 41-tone CS/PB. The only force strong enough to overcome
> the "closure" attained at 41 is a diamondic force.

Yes. But with a Miracle interpretation, the diamondic force isn't
required to work so hard, since there is only a partial closure at 41,
and total closure doesn't come until 72. i.e. Miracle will "give in"
to the diamond more easily and one doesn't need to invoke "variable
notes".
-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > > I guess what I want to know is, can Miracle-41 be considered a
> > > tempering of that PB, or is there no relationship between their
> > unison
> > > vectors.
> >
> > Is there a comprehensive way to find an answer to this question? I
> > can spit out the FPB for any set of unison vectors in no time . . .
> > please instruct (brain sugar low right now).
>
> I dunno. Is it enough to find a set of 11-limit unison vectors for
> Miracle-41 and see if they are linearly independent of those for
> your 41 tone PB.

How could they be? The latter span the entire 4D space. Anyway I'm not at the office anymore
... any FPBs will have to wait until tomorrow.

> I notice you use FPB for "Fokker periodicity block".
> Is there any other kind?

Yes. I keep Fokker in there only when the PB is a hyperparallelopiped, with one note in the
center. Other shapes, like most of Wilson's CS scales, are periodicity blocks too, but I drop
Fokker's name in general.

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
>
> He was aware he was "filling in the gaps" in his scale. Also the concept of Moments of
Symmetry
> and Constant Structure are general principles. Outside of being a straight linear temperment
> series I see no general principle in Miracle.

By the way, Kraig, it seems there is an even more general principle underlying all of this, that I
enunciated in my Hypothesis (and I'm sure Wilson understands in many unverbalized forms). In
a lattice with n primes, n-1 unison vectors will imply a certain generator, hence a certain narrow set
of linear temperaments. The generator also implies a certain set of Moments of Symmetry,
normally all Constant Structures. For example, in (3,5) space, the 81:80 unison vector alone
implies the 3:2 generator, meantone temperament, the pentatonic and diatonic scales (whether
tempered or CSs in JI), etc. MIRACLE is the generator and set of MOSs and CSs implied by
a certain set of superparticular unison vectors in (3,5,7,11) space . . . Whenever you "fill the
gaps" in a scale, you're naturally inclined to end up with a periodicity block, since a periodicity
block is as big a chunk of the lattice as possible given that you don't want any near-duplications
(which would be unison vectors, which define the periodicity block by delimiting it). We're
discussing all this in a lot more detail on the tuning-math list, in case you or anyone else is
interrested . . . You've been very dismissive of these ideas in the past, and if you can think
around them, fine, many different approaches may be mathematically equivalent, but to me the
Hypothesis cuts to the heart of what's going on with all this stuff.

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> Yes. But with a Miracle interpretation, the diamondic force isn't
> required to work so hard, since there is only a partial closure at 41,
> and total closure doesn't come until 72. i.e. Miracle will "give in"
> to the diamond more easily and one doesn't need to invoke "variable
> notes".

Don't see it. How does it give in more easily? You still need to go slightly beyond the partial
closure (a bit more so, in fact), so by this interpretation, perhaps Partch would have ended up
with 72 notes!?

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

> Yes. But with a Miracle interpretation, the diamondic force isn't
> required to work so hard, since there is only a partial closure at 41,
> and total closure doesn't come until 72. i.e. Miracle will "give in"
> to the diamond more easily and one doesn't need to invoke "variable
> notes".

I don't get this. Don't you need 45 notes in the MIRACLE chain to get Partch's scale? Aren't you
then invoking _four_, rather than two, variable notes, relative to the partial closure? Or are you
speaking of some of Partch's earlier scales from the _Exposition of Monophony_?

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> Dave!
> Please define what a second-class citizen is. You have a
harmonic construct (the diamond) and
> a melodic Construct (41) and a kink in their resolution. Its just a
reality

It has been assumed so until recently.

I assume the linear temperament corresponding to Wilson's
interpretation is the one that has an optimum fifth generator of
702.63 cents (or a fourth of 497.37c) giving a max 11-limit error of
8.7 cents. This generator has MOS at 1 2 (3) 5 (7) 12 (17) 29 41 (70
111) 152, (improper in parentheses). Notice that 53 doesn't make it in
there. Here's its mapping of primes to numbers of generators.

Prime No. gens (taking the fifth as the gen)
---------------
3 1
5 -8
7 -14
11 -18

This temperament disrespects the diamond by mapping 10:11 and 9:10 to
the same note. Both are best approximated by -10 generators. And one
has to do something ad-hoc (as Wilson did) to get them both onto a
keyboard based on this generator.

This is what I mean when I say it treats 10:11 (or I could have said
9:10) as a second-class citizen. Another way to put it is that the
11-limit diamond "overloads" this temperament.

This does not happen with Miracle temperament, or a Miracle keyboard.
-- Dave Keenan

Dave!
Somehow this didn't get sent earlier!
Paul had asked about mapping of the 13 limit diamond to 72, this one does at 53. 11/10 as a
second class citizen? no! Just because he maps it in a linear series in the opposite direction
doesn't mean it isn't as essential. I await a better mapping of the 13 limit diamond.

Dave Keenan wrote:

> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> > > Paul!
> > > Thought you would want to at least see this Starting on 6 with
> > the meat on 7 and 8.
> > > http://www.anaphoria.com/tres.PDF
> >
> > Thanks Kraig! Dave, do these change anything?
>
> How should they? On p6, I see 11/10 treated as a second-class citizen,
> this doesn't have to happen with a miracle keyboard mapping. What is
> the difference between p6 and p7? I don't see the relevance of p8.

-- Kraig Grady
North American Embassy of Anaphoria island
http://www.anaphoria.com

The Wandering Medicine Show
Wed. 8-9 KXLU 88.9 fm

--- In tuning@y..., Kraig Grady <kraiggrady@a...> wrote:
> Dave!
> Somehow this didn't get sent earlier!
> Paul had asked about mapping of the 13 limit diamond to 72,

I didn't ask but simply stated that there would be overlaps in such a mapping.

> this one does at 53.

With even more overlaps. I'm inclined to dislike this mapping since 53-tET is not consistent
through the 13-limit (or even the 11-limit).

--- In tuning@y..., "Jon Szanto" <JSZANTO@A...> wrote:
> > What do you mean by "change history"? Certainly I agree that Dave
> > may be overestimating the importance of MIRACLE in Partch's
> > thought.
>
> My point exactly, bucko! "MIRACLE in Partch's thought"? He wasn't
> thinking of it, unless you want to rewrite, or go back and find
> someone that said "You know, I showed HP this M scale and he
> went 'Holy mackeral, that's boffo!'".

Hey! Talk about misrepresenting me. Sheesh. I never once said Partch
was _thinking_ of Miracle. In fact I wrote:
"Harry Partch knew nothing of the Miracle temperament or the
corresponding keyboard ...". Please take more care, Paul and Jon.

> That's what I mean by change history, saying that Partch was
> unconsciously drawn to something that hadn't been written on or
> discussed.

I hope you understand that someone could be unconsciously drawn to
find a particular way of filling in the diamond gaps by how the
candidates for the additional notes _sound_ with the others. Nothing
to do with what anyone had written or discussed. But of course we can
only speculate, the evidence is sketchy, from the scales themselves,
and anecdotal evidence about HP's use of non-strict JI harmony.
-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> This temperament disrespects the diamond by mapping 10:11 and 9:10 to
> the same note. Both are best approximated by -10 generators.

Actually, Wilson approximates one by -10 generators, the other by 31 generators (IIRC).
There is no disrespect. By mapping them to the same key on the keyboard, Wilson
emphasizes the CS nature of the scale.
>
> This is what I mean when I say it treats 10:11 (or I could have said
> 9:10) as a second-class citizen. Another way to put it is that the
> 11-limit diamond "overloads" this temperament.

Only in the gentlest way possible.
>
> This does not happen with Miracle temperament, or a Miracle keyboard.

That's silly. It overloads MIRACLE-41 by _four_ notes, rather than the two notes by which it
overloads Wilson-41. And while it sits comfortably in MIRACLE-72, it would sit just as
comforably in a 72-tone scale generated by lots of other generators, such as the minor-third
(19/72 oct.) one that seemed to govern some of Wilson's 72-tone keyboard designs.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> >
> > Yes. But with a Miracle interpretation, the diamondic force isn't
> > required to work so hard, since there is only a partial closure at
41,
> > and total closure doesn't come until 72. i.e. Miracle will "give
in"
> > to the diamond more easily and one doesn't need to invoke
"variable
> > notes".
>
> Don't see it. How does it give in more easily? You still need to go
slightly beyond the partial
> closure (a bit more so, in fact), so by this interpretation, perhaps
Partch would have ended up
> with 72 notes!?

No he only had to go far enough to include the diamond (45 notes max).
The point is that with Miracle you _can_ go beyond the partial closure
to include 11/10 and 20/11, in addition to 10/9 and 9/5. With Wilsons
schismic interpretation you don't have that option. It doesn't matter
how many more notes you add to the chain, you will never get 11/10
separate from 10/9. The diamond overloads schismic-41.
-- Dave Keenan

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> > Yes. But with a Miracle interpretation, the diamondic force isn't
> > required to work so hard, since there is only a partial closure at
41,
> > and total closure doesn't come until 72. i.e. Miracle will "give
in"
> > to the diamond more easily and one doesn't need to invoke
"variable
> > notes".
>
> I don't get this. Don't you need 45 notes in the MIRACLE chain to
get Partch's scale?

Yes if you insist on a contiguous chain, but if you only want the
diamond plus maximal evenness, you only need 43 notes of Miracle.

> Aren't you
> then invoking _four_, rather than two, variable notes, relative to
> the partial closure?

I suppose you could say that, "relative to the partial closure". But I
don't think of these as "variable notes" because they are actually
generated as new notes by extending the chain. No amount of extending
a schismic fifth chain will generate 11/10 as a separate note from
10/9. So these really _must_ be considered as a variable note in the
schismic case.

> Or are you
> speaking of some of Partch's earlier scales from the _Exposition of
> Monophony_?

Yes.
-- Dave Keenan

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

> Yes if you insist on a contiguous chain, but if you only want the
> diamond plus maximal evenness, you only need 43 notes of Miracle.

What sort of "maximal evenness" are you referring to here?
>
> > Aren't you
> > then invoking _four_, rather than two, variable notes, relative to
> > the partial closure?
>
> I suppose you could say that, "relative to the partial closure". But I
> don't think of these as "variable notes" because they are actually
> generated as new notes by extending the chain. No amount of extending
> a schismic fifth chain will generate 11/10 as a separate note from
> 10/9.

I believe Kraig was saying that Wilson did extend the chain of fifths to do just that. And what
about the minor-third (19/72 oct.) generator? Or the major-third (23/72 oct.) generator? In those
cases you can certainly extend the chain to generate separate 11/10 and 10/9, just as in 72-tET.
>
> > Or are you
> > speaking of some of Partch's earlier scales from the _Exposition of
> > Monophony_?
>
> Yes.

You're speaking _only_ of those, or _also_ of those?

> ----- Original Message -----
> From: Dave Keenan <D.KEENAN@UQ.NET.AU>
> To: <tuning@yahoogroups.com>
> Sent: Monday, June 25, 2001 9:12 PM
> Subject: [tuning] Re: Partch's scales on the Miracle keyboard
>
>
> This is what I mean when I say it treats 10:11 (or I could have said
> 9:10) as a second-class citizen. Another way to put it is that the
> 11-limit diamond "overloads" this temperament.
>
> This does not happen with Miracle temperament, or a Miracle keyboard.

Hi Dave,

Thanks for such a clear explanation in response to Kraig. I wasn't
following what you meant by "overloaded"... now I understand.
I think that's a good term. Dictionary time...

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> >
> > This temperament disrespects the diamond by mapping 10:11 and 9:10
to
> > the same note. Both are best approximated by -10 generators.
>
> Actually, Wilson approximates one by -10 generators, the other by 31
generators (IIRC).

Ok. So I assumed the wrong linear temperament. It must correspond to
the one whose 11-limit optimum fifth is 702.193c giving a max 11-limit
error of 4.3c. It has MOS at 1 2 (3) 5 (7) 12 (17 29) 41 (53) 94,
(improper in paren.) and the following mapping.

Prime No. gens
---------------
3 1
5 -8
7 14
11 23

Well hey, you need a chain of 75 notes of this temperament in order to
fit the 11-limit diamond!

> There is no disrespect. By mapping them to the same key on the
keyboard, Wilson
> emphasizes the CS nature of the scale.

As I think Carl Lumma said: "CS is irrelevant for more than about 14
notes". CS is totally irrelevant for 41 or 43 notes. CS seems far more
important for those of its subsets that are of a size that can form a
melodic gestalt ("white note" scales).

> > This does not happen with Miracle temperament, or a Miracle
keyboard.
>
> That's silly. It overloads MIRACLE-41 by _four_ notes, rather than
the two notes by which it
> overloads Wilson-41. And while it sits comfortably in MIRACLE-72, it
would sit just as
> comforably in a 72-tone scale generated by lots of other generators,
such as the minor-third
> (19/72 oct.) one that seemed to govern some of Wilson's 72-tone
keyboard designs.

I said "Miracle temperament", not Miracle-41 or 72. But given the new
understanding of Wilson's temperament, there are no overoads in
it either. But (since you seem to think it important) its smallest MOS
to contain the diamond has 94 notes!

-- Dave Keenan

> ----- Original Message -----
> From: Paul Erlich <paul@stretch-music.com>
> To: <tuning@yahoogroups.com>
> Sent: Monday, June 25, 2001 8:56 PM
> Subject: [tuning] Re: Partch's scales on the Miracle keyboard
>
>
> By the way, Kraig, it seems there is an even more general
> principle underlying all of this, that I enunciated in my
> Hypothesis (and I'm sure Wilson understands in many
> unverbalized forms). In a lattice with n primes, n-1 unison
> vectors will imply a certain generator, hence a certain
> narrow set of linear temperaments. The generator also implies
> a certain set of Moments of Symmetry, normally all Constant
> Structures. ...

Paul, this is great. Can you give us a few links to posts
where this has been developed?... I'd like to assemble it into
a webpage.

-monz
http://www.monz.org
"All roads lead to n^0"

_________________________________________________________
Do You Yahoo!?
Get your free @yahoo.com address at http://mail.yahoo.com

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> > Yes if you insist on a contiguous chain, but if you only want the
> > diamond plus maximal evenness, you only need 43 notes of Miracle.
>
> What sort of "maximal evenness" are you referring to here?

Using the term loosely, to refer to what Partch was presumably aiming
for.

> And what
> about the minor-third (19/72 oct.) generator? Or the major-third
(23/72 oct.) generator?

You tell me.

> In those
> cases you can certainly extend the chain to generate separate 11/10
and 10/9, just as in 72-tET.

Ok But how long a chain do they take to fit the diamond, and at what
accuracy?

> > > Or are you
> > > speaking of some of Partch's earlier scales from the _Exposition
of
> > > Monophony_?
> >
> > Yes.
>
> You're speaking _only_ of those, or _also_ of those?

Also. But I'm prepared to fall back to "only", if necessary. :-)

-- Dave Keenan

--- In tuning@y..., "monz" <joemonz@y...> wrote:
> I wasn't
> following what you meant by "overloaded"... now I understand.
> I think that's a good term. Dictionary time...

Ok. But notice that I apparently picked on the wrong temperament,
according to Paul. The correct one isn't overloaded by the diamond, it
just has 75-29 = 46 holes in its diamond! compared to Miracle's 45-29
= 16 holes.

-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > And what
> > about the minor-third (19/72 oct.) generator? Or the major-third
> (23/72 oct.) generator?

316.74c gives 3.0c errors. MOS at 1 (2) 3 4 (7 11) 15 19 (34) 53 72
(125) 197. Mapping

Prime No. gens
---------------
3 6
5 5
7 22
11 -21

So it needs a chain of 87 notes for the diamond. 58 holes.

I can't find a suitable temperament with a major third generator.

-- Dave Keenan

I get it now. In http://www.anaphoria.com/tres.PDF
p6 uses the first schismic mapping that I assumed, which overloads two
keys.
p7 uses the second one that has no overloads but lots of holes, which
Wilson has filled using lots of notes that have nothing to do with
Partch's scale.

I think Secor's Miracle mapping is way better than either of these.

-- Dave Keenan

In-Reply-To: <9h8q7a+bik7@eGroups.com>
Jon Szanto wrote:

> ...or rather: Dave, Dave, Dave! That is just a bit cosmic for someone
> as pragmatic as Partch. *Everything* has been there forever, if you
> want to follow that line of reasoning. I'm just not buying the
> concept that now that people have codified (or are in the process of)
> the M stuff, that you can say past *musical* choices were guided by
> this... this... 'unknown-but-existing construct'.

Hang on there!

When we came up with the Miracle ideas, one of the objections aired on
this list was that they weren't implicit in existing music. By the time
meantone was theoretically formulated, there was already a body of
significant music that depended on it. So if Miracle's such a good idea,
it was said, why isn't it something composers were able to discover
without the theoretical structure?

Well, now people may or may not be suggesting that the concept underlies
Partch's thinking, that can't be right because the theory didn't exist!
Can the proponents of these extreme positions work it out amongst
themselves, and leave the rest of us to consider the historical and
acoustic facts?

Graham

In-Reply-To: <9h93vi+4mv0@eGroups.com>
Dave Keenan wrote:

> No he only had to go far enough to include the diamond (45 notes max).
> The point is that with Miracle you _can_ go beyond the partial closure
> to include 11/10 and 20/11, in addition to 10/9 and 9/5. With Wilsons
> schismic interpretation you don't have that option. It doesn't matter
> how many more notes you add to the chain, you will never get 11/10
> separate from 10/9. The diamond overloads schismic-41.

This looks exactly like what Wilson himself suggested in D'alessandro:
Like A Hurricane. You start with a keyboard to play an ET with some
duplicates either end of the keyboard, to give you a choice about
transposition. Then you switch to JI, but tune those duplicate keys
differently to get more notes.

Apply this technique to a 45 key Miracle keyboard, and you can either get
41-equal with duplicates or a superset of Partch's earlier 43 note scale.
OTOH, the schismic mapping Wilson used in Xenharmonikon needs a split
key for 11:10 and 10:9.

So now we can see that page 6 of <http://www.anaphoria.com/tres.PDF>
shows Wilson using a Miracle-like mapping to cover Partch's scale in
exactly this fashion. There are also no gaps in the 41-note subset, so
this is a better match, if it is a consistent temperament.

I say "Miracle-like" because it seems to be a solution to the problem of
fitting Miracle to a hexagonal keyboard. That is, the problem Kraig said
didn't exist.

If you look at page 7, one diagonal goes

1 16(15) 8 11 13
- --(--) - -- --
1 15(14) 7 9 10

So it's the Miracle generator extended to the 13-limit. It looks like
the only Miracle unison vector missing is that which would make two 11:9
neutral thirds equal to a 3:2 fifth.

Graham

In-Reply-To: <3B380BB5.B6668C2@anaphoria.com>
Kraig wrote:

> I await a better mapping of the 13
> limit diamond.

Two keyboards tuned to 29-equal, 16.4 cents apart.

Graham

--- In tuning@y..., graham@m... wrote:
> So now we can see that page 6 of <http://www.anaphoria.com/tres.PDF>
> shows Wilson using a Miracle-like mapping to cover Partch's scale in
> exactly this fashion. There are also no gaps in the 41-note subset,
so
> this is a better match, if it is a consistent temperament.
>
> I say "Miracle-like" because it seems to be a solution to the
problem of
> fitting Miracle to a hexagonal keyboard. That is, the problem Kraig
said
> didn't exist.
>
> If you look at page 7, one diagonal goes
>
> 1 16(15) 8 11 13
> - --(--) - -- --
> 1 15(14) 7 9 10
>
> So it's the Miracle generator extended to the 13-limit. It looks
like
> the only Miracle unison vector missing is that which would make two
11:9
> neutral thirds equal to a 3:2 fifth.

No. Page 6 and page 7 are two slightly different schismic mappings as
I explained earlier (but you probably hadn't got to it when you wrote
the above). These do not use Miracle generators. Of course you will
find short chains of apparently Miracle generators in schismic, just
as you will find short chains of fifths in Miracle.

I still don't see how to map Miracle to a hexagonal keyboard. Skewed
rectangular looks way better to me for Miracle.

-- Dave Keenan

In-Reply-To: <memo.756665@cix.compulink.co.uk>
I wrote:

> > I await a better mapping of the 13
> > limit diamond.
>
> Two keyboards tuned to 29-equal, 16.4 cents apart.

Or three keyboards tuned to the relevant just intervals if you want a
whole diamond. 87 notes required. On closer inspection, though, this
isn't unique. So I can't improve on Wilson's mapping.

Graham

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> Ok. So I assumed the wrong linear temperament. It must correspond
to
> the one whose 11-limit optimum fifth is 702.193c giving a max 11-
limit
> error of 4.3c. It has MOS at 1 2 (3) 5 (7) 12 (17 29) 41 (53) 94,
> (improper in paren.) and the following mapping.
>
> Prime No. gens
> ---------------
> 3 1
> 5 -8
> 7 14
> 11 23
>
> Well hey, you need a chain of 75 notes of this temperament in order
to
> fit the 11-limit diamond!

Well then that's not the right one . . .
>
> > There is no disrespect. By mapping them to the same key on the
> keyboard, Wilson
> > emphasizes the CS nature of the scale.
>
> As I think Carl Lumma said: "CS is irrelevant for more than about
14
> notes".

Bounce that one off Kraig!

--- In tuning@y..., "monz" <joemonz@y...> wrote:
>
> > ----- Original Message -----
> > From: Paul Erlich <paul@s...>
> > To: <tuning@y...>
> > Sent: Monday, June 25, 2001 8:56 PM
> > Subject: [tuning] Re: Partch's scales on the Miracle keyboard
> >
> >
> > By the way, Kraig, it seems there is an even more general
> > principle underlying all of this, that I enunciated in my
> > Hypothesis (and I'm sure Wilson understands in many
> > unverbalized forms). In a lattice with n primes, n-1 unison
> > vectors will imply a certain generator, hence a certain
> > narrow set of linear temperaments. The generator also implies
> > a certain set of Moments of Symmetry, normally all Constant
> > Structures. ...
>
>
> Paul, this is great. Can you give us a few links to posts
> where this has been developed?... I'd like to assemble it into
> a webpage.
>
It's all being discussed quite heavily on the tuning-math list. What
I didn't mention above was that each MOS corresponds to a choice of
an nth unison vector, treated as a "chromatic" unison vector (i.e.,
_not_ tempered out or treated as an equivalence).

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > > And what
> > > about the minor-third (19/72 oct.) generator? Or the major-
third
> > (23/72 oct.) generator?
>
> 316.74c gives 3.0c errors. MOS at 1 (2) 3 4 (7 11) 15 19 (34) 53 72
> (125) 197. Mapping
>
> Prime No. gens
> ---------------
> 3 6
> 5 5
> 7 22
> 11 -21
>
> So it needs a chain of 87 notes for the diamond.

Now that's just silly. The chain _closes_ after 72 notes if you use
the 19/72 oct. generator! But I don't think linear temperaments are
too relevant anyway.

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> I get it now. In http://www.anaphoria.com/tres.PDF
> p6 uses the first schismic mapping that I assumed, which overloads
two
> keys.
> p7 uses the second one that has no overloads but lots of holes,
which
> Wilson has filled using lots of notes that have nothing to do with
> Partch's scale.
>
> I think Secor's Miracle mapping is way better than either of these.
>
> -- Dave Keenan

Wouldn't you rather have 41 keys (with two of the diamonic ones
split) than 72 keys, if all you're going to be playing is Partch's 43?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > Prime No. gens
> > ---------------
> > 3 6
> > 5 5
> > 7 22
> > 11 -21
> >
> > So it needs a chain of 87 notes for the diamond.
>
> Now that's just silly. The chain _closes_ after 72 notes if you use
> the 19/72 oct. generator! But I don't think linear temperaments are
> too relevant anyway.

Of course linear temperaments are relevant, not any precise value of
the generator but a temperament's mapping from primes to numbers of
generators is what informs the keyboard layout. And the length of
chain required for the diamond determines how compact th escale is
on the keyboard.

But yes, it was silly of me to forget to treat this one modulo 72. So
it needs 72 notes for the diamond. Still not very good.
-- Dave Keenan

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> Wouldn't you rather have 41 keys (with two of the diamonic ones
> split) than 72 keys, if all you're going to be playing is Partch's
43?

[My earlier reply to this seems to have vanished without a trace. Bloody
Yahoo! So this time I'm writing it in my email program, not the web
browser, like I shoulda done in the first place.]

If this was the only choice then yes. But in fact the choice is between
Schismic-modulo-41 plus 2, versus Miracle-45.

I'm proposing the following Miracle keyboard mapping (appropriately skewed).

20/11 64/33
x 10/9 32/27 14/11 27/20 16/11 14/9 5/3 16/9 40/21
81/80 12/11 7/6 5/4 4/3 10/7 32/21 18/11 7/4 15/8
1/1 16/15 8/7 11/9 21/16 7/5 3/2 8/5 12/7 11/6
160/81 21/20 9/8 6/5 9/7 11/8 40/27 11/7 27/16 9/5
y 33/32 11/10

If desired, the gaps marked x and y can be filled with repeats of 33/32 and
64/33.

Of course 81/80 and 27/20 [I won't bother writing "and inverses" in future]
are not on the keys they would be on in an _open_ Miracle chain, but then
in Wilson's mapping 33/32, 11/10, 11/9, 11/8, 11/7 and 11/6 are not where
they would be for an _open_ schismic chain (which you can see on p6 of
http://www.anaphoria.com/tres.PDF

So I've used Miracle-modulo-41 for 81/80 and 27/20. Miracle-modulo-41 would
map 11/10 and 10/9 to a single key, just as Schismic-modulo-41 does. But
rather than splitting the key, or placing 11/10 on the closest available
spare key as Wilson appears to have done, in Miracle we can place it where
it would be in an open Miracle chain. This only takes us to 45 keys (2
holes), whereas in Schismic it would put 11/10 way out in the boondocks,
with 63 keys (20 holes).

Wilson's (p6) placement of 11/10 means that there is no consistent keyboard
pattern for all of the diamond hexads (of a given o/u). The diamond is
broken on this keyboard.

The above Miracle-45 mapping has the same pattern for _all_ the diamond
hexads (of a given o/u).

Also, if you take the JI patterns on the Schismic keyboard and play them
everywhere, you will end up with more or larger comma errors than doing the
same thing with the above Miracle-45 proposal.

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

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> Wouldn't you rather have 41 keys (with two of the diamonic ones
> split) than 72 keys, if all you're going to be playing is Partch's
43?

If that was the choice, then yes. For Partch's early 43, the choice is
between 41 keys with two split, versus 45 keys of Miracle. For his
latest 43 we're looking at 63 of Miracle (but later I reduce this
also to 45 keys).

I think split keys on a keyboard of this size and type are highly
impractical. It just isn't going to happen. What you're really saying
to Partch is "Sorry fella but you're going to have to either lose two
of those notes, or stick them outside the pattern, as on Wilson's
layout, p6)". This means that some of the diamond hexads will have a
_different_keyboard_pattern_ to the others (of the same o/u).

I think it's obvious that 45 of miracle is better for the early scale,
but its the later one that we're really interested in. Here's Secor's
Miracle mapping.
160/81
40/27 20/11 64/33
10/9 32/27 14/11 16/11 14/9 5/3 16/9 40/21
12/11 7/6 5/4 4/3 10/7 32/21 18/11 7/4 15/8
1/1 16/15 8/7 11/9 21/16 7/5 3/2 8/5 12/7 11/6
21/20 9/8 6/5 9/7 11/8 11/7 27/16 9/5
33/32 11/10 27/20
81/80

But if you think its ok to consider Kleismic modulo 72, even though
the diamond requires 87... In other words, if some notes can be out of
their logical place with regard to the JI mapping ... Then I could do
the same thing with Partch's latest 43 on Miracle, and only require 45
keys for that too. i.e. put 81/80 where 49/40 was in the early scale,
and put 27/20 where 15/11 was (I won't bother writing "and
correspondingly for their inversions" any more).

20/11 64/33
x 10/9 32/27 14/11 27/20 16/11 14/9 5/3 16/9 40/21
81/80 12/11 7/6 5/4 4/3 10/7 32/21 18/11 7/4 15/8
1/1 16/15 8/7 11/9 21/16 7/5 3/2 8/5 12/7 11/6
160/81 21/20 9/8 6/5 9/7 11/8 40/27 11/7 27/16 9/5
y 33/32 11/10

If desired, one could fill the two holes marked x and y with copies of
33/32 and 64/33 respectively.

In other words I'm doing Miracle modulo 41 with a twist. Like Wilsons
(p6) mapping, strict Miracle modulo 41 would cause 11/10 to have to
share with 10/9. It would also move 33/32 to the hole that might be
considered to be for 28/27. But instead of doing like Wilson's p6
mapping and just putting 11/10 in the nearest available position, I do
like his p7 mapping and put it where it should be for a non-mod-41
mapping. In Wilson's case this puts 11/10 way out in the boondocks,
extending the mapping to 63 notes (20 holes). In the Miracle case it
only extends it to 45 notes (2 holes).

The above Miracle-45 keyboard mapping gives the same keyboard pattern
for all of the diamond hexads.

Wilson's mappings either break up the diamond (p6) or require too many
keys (p7). Miracle-45 doesn't break up the diamond and doesn't have
too many keys, but it does have non-standard patterns for
strictly-just chords involving the non-diamond notes 81/80 and 27/20
(and their inverses).

So it's a question of what's more important for a Partch keyboard
mapping to regularise, the diamond, or the chains of just fifths?

-- Dave Keenan

Ok. So the post I thought I'd lost, turns up hours later, after I've
retyped it. Ho hum. Sorry about the duplication.

Check out these more realistic diagrams of the mapping of Partch's 43
note scale to Miracle-45 keyboards, both hexagonal and square
(actually slightly rhombic).

http://dkeenan.com/Music/Miracle/Miracle45Keyboard.doc

For those who can't read MsWord files I've included a GIF at
http://dkeenan.com/Music/Miracle/Miracle45Keyboard.gif
but it's a bit messy.

-- Dave Keenan

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

> Of course linear temperaments are relevant, not any precise value of
> the generator but a temperament's mapping from primes to numbers of
> generators is what informs the keyboard layout.

Why must it necessarily be a
particular linear temperament?
Nothing is actually _tempered_ on
these keyboard layouts.

--- In tuning@y..., "Dave Keenan"
<D.KEENAN@U...> wrote:
> Ok. So the post I thought I'd lost, turns up hours later, after I've
> retyped it. Ho hum. Sorry about the duplication.
>
> Check out these more realistic diagrams of the mapping of Partch's 43
> note scale to Miracle-45 keyboards, both hexagonal and square
> (actually slightly rhombic).
>
> http://dkeenan.com/Music/Miracle/Miracle45Keyboard.doc
>
> For those who can't read MsWord files I've included a GIF at
> http://dkeenan.com/Music/Miracle/Miracle45Keyboard.gif
> but it's a bit messy.
>
> -- Dave Keenan

How about a version
corresponding with the earlier
_Exposition of Monophony_ scale,
that will be truly consistent with
the MIRACLE mapping?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan"
> <D.KEENAN@U...> wrote:
>
> > Of course linear temperaments are relevant, not any precise value
of
> > the generator but a temperament's mapping from primes to numbers
of
> > generators is what informs the keyboard layout.
>
> Why must it necessarily be a
> particular linear temperament?
> Nothing is actually _tempered_ on
> these keyboard layouts.

Because a keyboard is only 2D it can only directly represent a linear
temperament (which is 2D when considerd octave-specifically).
Everything else, it can only represent approximately.

To prove that I know whereof I speak, I have created, for your
enjoyment, an automatic keyboard mapper for _any_ linear temperament
(octave-based or otherwise) up to a maximum of 72 notes per octave.
You can even see the mapping change in real time, as you change the
generator.

It is a 220k Excel spreadsheet. The only problem with it is that you
will need to rotate your computer monitor 90 degrees clockwise (or
turn your head the other way). :-)

http://dkeenan.com/Music/KeyboardMapper.xls

Regards,
-- Dave Keenan

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> How about a version [of the Miracle keyboard mapping]
> corresponding with the earlier
> _Exposition of Monophony_ scale,
> that will be truly consistent with
> the MIRACLE mapping?

You can do it yourself if you have MsWord. I already gave it in ASCII
form in the post that started this thread.

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., "Dave Keenan"
> > <D.KEENAN@U...> wrote:
> >
> > > Of course linear temperaments are relevant, not any precise
value
> of
> > > the generator but a temperament's mapping from primes to
numbers
> of
> > > generators is what informs the keyboard layout.
> >
> > Why must it necessarily be a
> > particular linear temperament?
> > Nothing is actually _tempered_ on
> > these keyboard layouts.
>
> Because a keyboard is only 2D it can only directly represent a
linear
> temperament (which is 2D when considerd octave-specifically).
> Everything else, it can only represent approximately.

That's a given. I still don't see why Wilson's keyboard designs must
be understood in terms of an abstract (i.e., not audibly realized)
linear temperament.
>
> To prove that I know whereof I speak, I have created, for your
> enjoyment, an automatic keyboard mapper for _any_ linear
temperament
> (octave-based or otherwise) up to a maximum of 72 notes per octave.
> You can even see the mapping change in real time, as you change the
> generator.
>
> It is a 220k Excel spreadsheet. The only problem with it is that
you
> will need to rotate your computer monitor 90 degrees clockwise (or
> turn your head the other way). :-)
>
> http://dkeenan.com/Music/KeyboardMapper.xls
>
Cool! I notice a bug -- when I type in period 600, generator 709.09,
11 generators per chain, the "273" and "982" labels are missing from
their corresponding points. Also there's a bunch of numerical text at
the top which is all garbled.

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > How about a version [of the Miracle keyboard mapping]
> > corresponding with the earlier
> > _Exposition of Monophony_ scale,
> > that will be truly consistent with
> > the MIRACLE mapping?
>
> You can do it yourself if you have MsWord. I already gave it in
ASCII
> form in the post that started this thread.

I just meant that if you were to make this webpage into an article,
for XH say, you might want to show how Secor's mapping leads to a
much more compact arrangement when the earlier version of Partch's
scale is used in place of the "standard" one.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > > --- In tuning@y..., "Dave Keenan"
> > > <D.KEENAN@U...> wrote:
> > >
> > > > Of course linear temperaments are relevant, not any precise
> value
> > of
> > > > the generator but a temperament's mapping from primes to
> numbers
> > of
> > > > generators is what informs the keyboard layout.
> > >
> > > Why must it necessarily be a
> > > particular linear temperament?
> > > Nothing is actually _tempered_ on
> > > these keyboard layouts.
> >
> > Because a keyboard is only 2D it can only directly represent a
> linear
> > temperament (which is 2D when considerd octave-specifically).
> > Everything else, it can only represent approximately.
>
> That's a given. I still don't see why Wilson's keyboard designs must
> be understood in terms of an abstract (i.e., not audibly realized)
> linear temperament.

Because, a chord pattern will work in every position for the
temperament. It won't work in every position for the JI scale. But the
_size_ and _quantity_ of the _errors_ you get in the available "false"
chords for the JI scale, depends entirely on which linear temperament
your keyboard is based on.

This is one reason why I find a Miracle-45 keyboard to be more
suitable for both of Partch's 43 note scales, than Schismic-41-plus-2.

> > http://dkeenan.com/Music/KeyboardMapper.xls
> >
> Cool! I notice a bug -- when I type in period 600, generator 709.09,
> 11 generators per chain, the "273" and "982" labels are missing from
> their corresponding points.

Thanks. That's fixed now at the above URL.

> Also there's a bunch of numerical text at
> the top which is all garbled.

Yeah. It seems like a bug in Excel to me. Ignore it. It's only the
data labels (cents values) for all the points (keys) that don't make
it into the chart range. You can change the Y-axis scale to show you
two octaves or whatever.

I also had a request for a version for older Excels. I did a "Save as
Microsoft Excel 5.0/95 workbook". The only thing I can see that it
lost in the translation was the larger circles I had for the keys.
They are now just little dots.
It's at http://dkeenan.com/Music/KeyboardMapper5.xls

-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > > --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > > > --- In tuning@y..., "Dave Keenan"
> > > > <D.KEENAN@U...> wrote:
> > > >
> > > > > Of course linear temperaments are relevant, not any precise
> > value
> > > of
> > > > > the generator but a temperament's mapping from primes to
> > numbers
> > > of
> > > > > generators is what informs the keyboard layout.
> > > >
> > > > Why must it necessarily be a
> > > > particular linear temperament?
> > > > Nothing is actually _tempered_ on
> > > > these keyboard layouts.
> > >
> > > Because a keyboard is only 2D it can only directly represent a
> > linear
> > > temperament (which is 2D when considerd octave-specifically).
> > > Everything else, it can only represent approximately.
> >
> > That's a given. I still don't see why Wilson's keyboard designs must
> > be understood in terms of an abstract (i.e., not audibly realized)
> > linear temperament.
>
> Because, a chord pattern will work in every position for the
> temperament. It won't work in every position for the JI scale. But the
> _size_ and _quantity_ of the _errors_ you get in the available "false"
> chords for the JI scale, depends entirely on which linear temperament
> your keyboard is based on.

Not if it's not based on a linear temperament! Arrgh! Why are we talking past each other here?
>
> This is one reason why I find a Miracle-45 keyboard to be more
> suitable for both of Partch's 43 note scales, than Schismic-41-plus-2.

But for the _Genesis_ scale, you're mixing a bit of a "outside" linear temperament to get the
notes within the chain of 45 generators, right? So that mapping is not really based on a single
fixed linear temperament . . . ?

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>>> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>>>> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>>>>> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>>>>>> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>>>>>>> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>>>>>>>> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>>>>>>>>> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>>>>>>>>>> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>>>>>>>>>>> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>>>>>>>>>>>> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
>>>>>>>>>>>>
>>>>>>>>>>>> Why are we talking past each other here?

Thing not said may be thing not heard.

It shore makes purty colors in the e mailer.
Cept for most o them is purple...

They wrote they wrote into a purple moat.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> Not if it's not based on a linear temperament! Arrgh! Why are we
talking past each other here?

Ok, so how do you make a 2D keyboard mappping for a 3D (e.g. 5-limit)
or 4D (e.g. 7-limit) or 5D (e.g. 11-limit) JI scale, that has
(a) pitch monotonically increasing from left to right,
(b) constant patterns for all (strict) JI chords,
(c) no pitches appear on more than one key,
_without_ it corresponding to some linear temperament? A single
example would be fine.

> But for the _Genesis_ scale, you're mixing a bit of a "outside"
linear temperament to get the
> notes within the chain of 45 generators, right? So that mapping is
not really based on a single
> fixed linear temperament . . . ?

Yes it's dirty, but I figure it's not as bad as Wilson's ad-hoc extra
key for 11/10 because it doesn't affect any notes of the diamond.

-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > Not if it's not based on a linear temperament! Arrgh! Why are we
> talking past each other here?
>
> Ok, so how do you make a 2D keyboard mappping for a 3D (e.g. 5-
limit)
> or 4D (e.g. 7-limit) or 5D (e.g. 11-limit) JI scale, that has
> (a) pitch monotonically increasing from left to right,
> (b) constant patterns for all (strict) JI chords,
> (c) no pitches appear on more than one key,
> _without_ it corresponding to some linear temperament? A single
> example would be fine.

Wilson's mappings don't always satisfy all these criteria; therefore
it would be wrong to say that each of his mappings implies a fixed
linear temperament (which was my only point). Wilson's creations are
based on deep principles which you and I probably know very little
about. Not to say we can't just proceed on our own.
>
> > But for the _Genesis_ scale, you're mixing a bit of a "outside"
> linear temperament to get the
> > notes within the chain of 45 generators, right? So that mapping
is
> not really based on a single
> > fixed linear temperament . . . ?
>
> Yes it's dirty, but I figure it's not as bad as Wilson's ad-hoc
extra
> key for 11/10 because it doesn't affect any notes of the diamond.
>
> -- Dave Keenan

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > > Not if it's not based on a linear temperament! Arrgh! Why are we
> > talking past each other here?
> >
> > Ok, so how do you make a 2D keyboard mappping for a 3D (e.g. 5-
> limit)
> > or 4D (e.g. 7-limit) or 5D (e.g. 11-limit) JI scale, that has
> > (a) pitch monotonically increasing from left to right,
> > (b) constant patterns for all (strict) JI chords,
> > (c) no pitches appear on more than one key,
> > _without_ it corresponding to some linear temperament? A single
> > example would be fine.
>
> Wilson's mappings don't always satisfy all these criteria; therefore
> it would be wrong to say that each of his mappings implies a fixed
> linear temperament (which was my only point). Wilson's creations are
> based on deep principles which you and I probably know very little
> about. Not to say we can't just proceed on our own.

Ooh. What a lovely cop-out! :-)

So you agree that one can't satisfy (a), (b) and (c) above on a 2D
keyboard without it corresponding to a particular linear temperamant?

So point me to one of these non-linear Wilson keyboard layouts so we
can see if we can possibly figure out why they violate one or more of
these conditions.

-- Dave Keenan

--- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
> > > --- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> > > > Not if it's not based on a linear temperament! Arrgh! Why are we
> > > talking past each other here?
> > >
> > > Ok, so how do you make a 2D keyboard mappping for a 3D (e.g. 5-
> > limit)
> > > or 4D (e.g. 7-limit) or 5D (e.g. 11-limit) JI scale, that has
> > > (a) pitch monotonically increasing from left to right,
> > > (b) constant patterns for all (strict) JI chords,
> > > (c) no pitches appear on more than one key,
> > > _without_ it corresponding to some linear temperament? A single
> > > example would be fine.
> >
> > Wilson's mappings don't always satisfy all these criteria; therefore
> > it would be wrong to say that each of his mappings implies a fixed
> > linear temperament (which was my only point). Wilson's creations are
> > based on deep principles which you and I probably know very little
> > about. Not to say we can't just proceed on our own.
>
> Ooh. What a lovely cop-out! :-)

Actually, you took me in. The logical flaw is that, while some of Wilson's mappings may
correspond to linear temperaments, that in no way implies that the properties of those
temperaments are relevant when applied back to the mapping. The implication only flows in
one direction, since the abstraction in terms of a linear temperament does not necessarily capture
the logic of what Wilson did.

Also, a 41-CS mapping onto hexagonal keys will imply at least 20 different generators!
>
> So you agree that one can't satisfy (a), (b) and (c) above on a 2D
> keyboard without it corresponding to a particular linear temperamant?

Actually, it seems to me now that in some cases, many different linear temperaments will be
implied . . .

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> Actually, you took me in. The logical flaw is that, while some of
Wilson's mappings may
> correspond to linear temperaments, that in no way implies that the
properties of those
> temperaments are relevant when applied back to the mapping.

Maybe it doesn't imply it. But I'm still saying that it is so, and
still waiting for a counter-example.

> The
implication only flows in
> one direction, since the abstraction in terms of a linear
temperament does not necessarily capture
> the logic of what Wilson did.
>
> Also, a 41-CS mapping onto hexagonal keys will imply at least 20
different generators!

If that is so, then there are 20 different mappings, each one implying
a different linear temperament. I think a single keyboard mapping will
only imply a single linear temperament (i.e. a single mapping from
primes to numbers of generators and periods). Please give a
counter-example.

-- Dave Keenan

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

>I think a single keyboard mapping will
> only imply a single linear temperament (i.e. a single mapping from
> primes to numbers of generators and periods). Please give a
> counter-example.

You said Wilson's Partch keyboard implies one particular schismic-fifth-generated temperament,
then switched to another. But why wouldn't _any_ approximate 41-tET interval do? They would
all close after 41 repetitions, generating 41 keyboard positions in the process. Perhaps you are
focusing on the fact that the vertical positions of the fifths wrap around the keyboard (vertically)
only once, while that of any other (non-equivalent) interval wraps around more times (still
generating the same 41 keyboard positions in the process). But that is purely a _keyboard
layout_ issue, not a _tuning issue_. If you focus on one aspect of the keyboard mapping, and
connect that to a way of deriving a unique linear temperament from it, that's fine . . . but you can't
then turn that around and say that properties of that linear temperament, as opposed to those of
others also compatible with the mapping, are in some way more important in the context of the
mapping.

Wilson likes to use the fifth as the slowest-wrapping interval on the keyboard so that diatonic
scales would be fingered as similarly as possible to a familiar, Bosanquet- (and ultimately,
Halberstadt-) derived pattern. That's the only significance I see of that convention. For Partch
keyboards, the template for a hexad is clearly shown by WIlson -- wrapping will clearly be
involved when transposing the pattern by a large number of fifths. When the MIRACLE
generator, instead, becomes the slowest-wrapping interval, the hexad has its minimal vertical
extent and wrapping is rarely an issue. So yes, the MIRACLE-based keyboard may be
superior for Partch's music -- IF you're willing to give up familiar diatonic patterns -- but that says
nothing about what Partch was thinking, or what Wilson was not thinking.

--- In tuning@y..., "Paul Erlich" <paul@s...> wrote:
> --- In tuning@y..., "Dave Keenan" <D.KEENAN@U...> wrote:
>
> >I think a single keyboard mapping will
> > only imply a single linear temperament (i.e. a single mapping from
> > primes to numbers of generators and periods). Please give a
> > counter-example.
>
> You said Wilson's Partch keyboard implies one particular
schismic-fifth-generated temperament,
> then switched to another.

In the end, I said the p6 keyboard was one temperament and the p7
keyboard was the other (I did have a typo in there somewhere where I
wrote p6 when it should have been p7). So no, this isn't a
counterexample to my assertion. This was what caused it to dawn on me.
Sure they both had fifth generators, but they had different mappings
from primes to numbers of generators. Sorry if I didn't make that
clear.

> But why wouldn't _any_ approximate 41-tET interval do?

Because they will each give a different keyboard layout, with a
different set of 41-tET hexads not conforming to the same pattern as
all the others (i.e. wrapping around, as you put it). Try it yourself
in my spreadsheet.

> They would
> all close after 41 repetitions, generating 41 keyboard positions in
the process.

Sure, since you're talking about 41-tET (which has lower
dimensionality than the keyboard). But we were talking about JI scales
(with higer dimensionality than the keyboard) their best keyboard
layout will not necessarily close after 41, as Partch's does not in
Miracle.

> Perhaps you are
> focusing on the fact that the vertical positions of the fifths wrap
around the keyboard (vertically)
> only once, while that of any other (non-equivalent) interval wraps
around more times (still
> generating the same 41 keyboard positions in the process).

You bet I'm focussing on that!

> But that
is purely a _keyboard
> layout_ issue, not a _tuning issue_.

Huh? I thought we'd been talking about keyboard layout the whole time.

> If you focus on one aspect of
the keyboard mapping, and
> connect that to a way of deriving a unique linear temperament from
it, that's fine . . .

I'm glad you at last agree that there is a one-to-one relationship
between keyboard layouts and linear temperaments for these kinds of
keyboards.

One aspect? It just happens to be the most important aspect; the one
that determines which chords fit a constant pattern and which ones
don't.

> but you can't
> then turn that around and say that properties of that linear
temperament, as opposed to those of
> others also compatible with the mapping, are in some way more
important in the context of the
> mapping.

What others compatible with what mapping?

Is there a difference between a keyboard mapping and a keyboard
layout in your usage? I've been assuming they are the same thing.

Wilson describes his keyboard layouts as 31-like or 41-like. There is
no such thing. He is assuming we know that he means "like a mapping
based on a 31-tET or 41-tET _fifth_ generator. But to say "like
41-tET" is only to narrow it down to 20 completely different keyboard
layouts.

> Wilson likes to use the fifth as the slowest-wrapping interval on
the keyboard so that diatonic
> scales would be fingered as similarly as possible to a familiar,
Bosanquet- (and ultimately,
> Halberstadt-) derived pattern.

Sure.

> That's the only significance I see of
that convention. For Partch
> keyboards, the template for a hexad is clearly shown by WIlson --
wrapping will clearly be
> involved when transposing the pattern by a large number of fifths.
When the MIRACLE
> generator, instead, becomes the slowest-wrapping interval, the hexad
has its minimal vertical
> extent and wrapping is rarely an issue. So yes, the MIRACLE-based
keyboard may be
> superior for Partch's music -- IF you're willing to give up familiar
diatonic patterns --

Thank you. This was my whole point.

> but that says
> nothing about what Partch was thinking, or what Wilson was not
thinking.

What did I say Partch was thinking?

What did I say Wilson was not thinking?

-- Dave Keenan

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

> > If you focus on one aspect of
> the keyboard mapping, and
> > connect that to a way of deriving a unique linear temperament
from
> it, that's fine . . .
>
> I'm glad you at last agree that there is a one-to-one relationship
> between keyboard layouts and linear temperaments for these kinds of
> keyboards.
>
> One aspect? It just happens to be the most important aspect; the
one
> that determines which chords fit a constant pattern and which ones
> don't.

That's only _necessarily_ true if the linear temperament you've
assumed is in fact the tuning of the keyboard.
>
> > but you can't
> > then turn that around and say that properties of that linear
> temperament, as opposed to those of
> > others also compatible with the mapping, are in some way more
> important in the context of the
> > mapping.
>
> What others compatible with what mapping?

That should have been clear from what I wrote. A 41-tone CS can be
thought of in terms of 20 different generators.

Anyway, looks like our disagreements may have been purely verbal, so
no need to go back and analyze one another's language now . . . on to
better things.