Some alternative Halberstadt layouts

I started investigating how to generalize the Halberstadt layout for other
tunings besides 12. While there is still obviously a case to be made for
transpositional-invariant keyboards, it is my opinion that
transpositionally-noninvariant keyboards should be explored as well. I think
that the Halberstadt, in particular, has a number of distinct advantages
that aren't usually discussed round these parts:

1) It makes easily accessible certain prioritized scales, which can be of
great use for people who are just learning the instrument. The natural way
to arrange the keys are in terms of two complementary MOS's, since the MOS's
of any tuning add to form another MOS. However, this is not a requirement.
2) It makes certain keys easier to play than others, which is either a bug
or a feature, depending on your preference. Thus alternative Halberstadts
may be a natural setup for unequal or circulating temperaments.
3) It will probably be somewhat intuitive for existing keyboard players to
make the switch, as the proprioceptive cues that they are accustomed to can
also apply here.
4) They prioritize melodic relationships in such a way that certain
generalized keyboards do not. 1 out of 31 on an Axis harmonic table layout,
for example, is going to be much further away than whatever 5/4 is mapped
to. This means that while Halberstadts aren't so great for ease of playing
harmony, they may offer some advantages for melodic playing and soloing.

So while the Bosanquet keyboard is akin to, say, Esperanto, the notion of
extending the Halberstadt layout is more like Interlingua: it is not
designed to be "tuning neutral" or anything like that, but rather to pick up
on and emphasize the cues that existing keyboard players have.

The problem is that once you get much past 12, the Halberstadt layout
quickly becomes impractical if only two tiers are involved. 19-equal, for
example, lends itself to a 12+7 layout for prioritizing meantone. This makes
it pretty difficult to hit an octave. The solution, as I see it, is to come
up with a three-tiered Halberstadt system.

This is far from a complete analysis, but here are some ideas on how to
extend the Halberstadt concept to a three-tiered system. Instead of going
with complementary MOS's, I went with 22-tet's complementary pentachordal
major + superpyth[7] + superpyth[5] combo, which combine to form the whole
22-tet set (thanks Paul Erlich). The basic imprint for how the keys are laid
out is this (Fixed width font plz:)

| | | | | | | <-- superpyth[5]
| | | | | | | | | | | | | | <-- SPM
| | || | | || | | <-- superpyth[7]

or

| | | | | | | <-- superpyth[5]
| | || | | || | | <-- superpyth[7]
| | | | | | | | | | | | | | <-- SPM

or

| | | | | | | | | | | | | | <-- SPM
| | | | | | | <-- superpyth[5]
| | || | | || | | <-- superpyth[7]

etc.

In contrast, 12-equal is obviously this:

| | | | | <-- meantone[5]
| | || | | |<-- meantone[7]

So the basic essence, of the Halberstadt, as I see it, is come up with an
"imprint" for a tuning like the above, and find a way to come up with
irregularly shaped keys that tesselate to fill the whole plane. The 12-tet
approach centers around turning the white keys into keys with "pads" at the
bottom, and the black keys as "stubs" with no pad. This is in opposition to
the Wilson/Fokker/Carlos layout, where the keys are uniformly long and
slender. It also seems that any layout you come up with like this can be
transformed into a transpositionally-invariant hexagonal version, although
that's something I personally haven't worked out yet.

The question becomes, then, how to set the keys up irregularly to accomplish
this objective. I have no idea, so here is an incomplete list of different
options that I've been screwing around with so far. All of these have
stupidly irregular key widths because ASCII art is hard. Just imagine
everything is in nice proportion for all three tiers.

**22-tet - superpyth[5] on top, SPM in the middle, superpyth[7] on bottom**

A self-similar "fractal"-based approach:

│ │ │ │ ││ │ │ │ │ │ │ │ ││ │ │ ││ │ │ │ │
│ │ │ │ ││ │ │ │ │ │ │ │ ││ │ │ ││ │ │ │ │
│ │ │ │ ││ │ │ │ │ │ │ │ ││ │ │ ││ │ │ │ │
│ │ │ │ ││ │ │ │ │ │ │ │ ││ │ │ ││ │ │ │ │
│ │ │ │ ││ │ │ │ │ │ │ │ ││ │ │ ││ │ │ │ │
│ │ └┬┘ ││ └┬┘ │ │ │ └┬┘ ││ └┬┘ ││ └┬┘ │ │
│ │ │ ││ │ │ │ │ │ ││ │ ││ │ │ │
│ │ │ ││ │ │ │ │ │ ││ │ ││ │ │ │
│ └──┴┬─┘└─┬┴──┘ │ └──┴┬─┘└──┼──┘└─┬┴──┘ │
│ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │
└─────┴────┴─────┴─────┴─────┴─────┴─────┘

**22-tet - SPM on top, superpyth[5] in the middle, superpyth[7] on bottom**

A different fractal-based approach: The black keys look like "stubs" to the
white keys below, but they look like "pads" for the third tier above.

│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ └┬┘ └┬┘ └┬┘ └┬┘ │ └┬┘ └┬┘ └┬┘ └┬┘ └┬┘ └┬┘ │
│ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │
│ └─┬─┘ └─┬─┘ │ └─┬─┘ └─┬─┘ └─┬─┘ │
│ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │
└─────┴───────┴─────┴─────┴───────┴───────┴─────┘

**22-tet - superpyth[5] on top, superpyth[7] in the middle, SPM on bottom!**

The "different length pads" version. This one makes it rather easy to play a
22-tet "chromatic" scale by simply placing the hand on the middle rank and
firing away.

│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │
│ └┬┘ │ │ └┬┘ │ │ │ └┬┘ │ │ └┬┘ │ │ └┬┘ │ │ │ └┬┘ │ │ └┬┘ │ │ │ └┬┘ │ │ └┬┘
│ │ └┬┘ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │
│ │ └┬┘ │ └┬┴┬┘ │ └┬┘ │ └┬┘ │ └┬┴┬┘ │ └┬┘ │ └┬┴┬┘ │ └┬┘ │
└┬┘ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │
└──┴───┴───┴───┘ └───┴───┴───┴───┴───┴───┘ └───┴───┴───┴───┘
└───┴───┴───┴───┴───┴──┘

A version similar to the above, but with the superpyth[7] keys wider as
well:

│ │ ││ ││ ││ │ ││ ││ ││ ││ ││ ││ │ ││ ││ ││ ││ │ ││ ││ ││ ││
││ │ │
│ │ ││ ││ ││ │ ││ ││ ││ ││ ││ ││ │ ││ ││ ││ ││ │ ││ ││ ││ ││
││ │ │
│ │ ││ ││ ││ │ ││ ││ ││ ││ ││ ││ │ ││ ││ ││ ││ │ ││ ││ ││ ││
││ │ │
│ │ ││ ││ ││ │ ││ ││ ││ ││ ││ ││ │ ││ ││ ││ ││ │ ││ ││ ││ ││
││ │ │
│ └┬┘│ │└┬┘│ │ │└┬┘│ │└┬┘│ │└┬┘│ │ │└┬┘│ │└┬┘│ │ │└┬┘│ │└┬┘│
│└┬┘ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │
│ │ └─┬─┘ │ └─┬┴┬─┘ │ └─┬─┘ │ └─┬─┘ │ └─┬┴┬─┘ │ └─┬─┘ │ └─┬┴┬─┘ │ └─┬─┘ │
└─┬─┘ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │
└──┴───┴───┴───┘ └───┴───┴───┴───┴───┴───┘ └───┴───┴───┴───┘
└───┴───┴───┴───┴───┴──┘

A kind of hybrid fractal version similar to the above, but with the
superpyth[7] keys wider as pads rather than stubs. This probably is the
easiest to play a chromatic scale on:

│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │
│ └┬┘┌┘ └┐└┬┘┌┘ │ └┐└┬┘┌┘ └┐└┬┘┌┘ └┐└┬┘┌┘ │ └┐└┬┘┌┘ └┐└┬┘┌┘ │ └┐└┬┘┌┘
└┐└┬┘┌┘ └┐└┬┘ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │
│ │ └─┬─┘ │ └─┬┴┬─┘ │ └─┬─┘ │ └─┬─┘ │ └─┬┴┬─┘ │ └─┬─┘ │ └─┬┴┬─┘ │ └─┬─┘ │
└─┬─┘ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │
└──┴───┴───┴───┘ └───┴───┴───┴───┴───┴───┘ └───┴───┴───┴───┘
└───┴───┴───┴───┴───┴──┘

For both of these you could also eliminate the gap on the SPM tier by making
the pads wider, but I found it unintuitive since I find it indicates a cue
that the notes adjacent in pitch. YMMV.

Regular way to set up 12-equal:

│ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │
│ └─┬─┘ └─┬─┘ │ └─┬─┘ └─┬─┘ └─┬─┘ │
│ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │
└─────┴─────┴─────┴─────┴─────┴─────┴─────┘

Alternate 12-equal that has 5 as the white keys and 7 as the black keys. The
white key widths are distorted and severely out of proportion because I hate
ASCII art:

│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
├──┘ └─┬─┘ └─┬─┘ └──┼──┘ └─┬─┘ └──┼──┘ └─┬─┘ └─┬─┘ └──┼──┘ └─┬─┘ └──┤
│ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │
└──────┴─────┴──────┴──────┴──────┴──────┴─────┴──────┴──────┴──────┘

A pseudo-jankoized version of the above, in which the black keys become pads
as well:

│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ └┐ ┌┘ └┐ ┌┘ └┐ ┌┘ │ └┐ ┌┘ └┐ ┌┘ │ └┐ ┌┘ └┐ ┌┘ └┐ ┌┘ │ └┐ ┌┘ └┐ ┌┘ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
├──┘ └─┬─┘ └─┬─┘ └──┼──┘ └─┬─┘ └──┼──┘ └─┬─┘ └─┬─┘ └──┼──┘ └─┬─┘ └──┤
│ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │
└──────┴─────┴──────┴──────┴──────┴──────┴─────┴──────┴──────┴──────┘

There are a ton of ways to set things like this up - have fun with it! If
anyone knows any further work that's been done on this type of thing, please
post it up as well. I see that Erv Wilson has a lot of writing on the
subject.

-Mike

Mike Battaglia <battaglia01@...> wrote:
> I started investigating how to generalize the Halberstadt
> layout for other tunings besides 12. While there is still
> obviously a case to be made for transpositional-invariant
> keyboards, it is my opinion that
> transpositionally-noninvariant keyboards should be
> explored as well. I think that the Halberstadt, in
> particular, has a number of distinct advantages that
> aren't usually discussed round these parts:

There's a survey of keyboards in the special microtonal
edition of Computer Music Journal, from the 80s sometime.
That uses the word "accretion" for this class of
keyboards. You'll have to check the article to see exactly
what term was used.

As well as the advantages you give, note that
accretion-type keyboards tend to be a better fit to
notation than generalized ones. I don't think I've seen a
notation that isn't based around a scale of nominals, which
would map to either the black or white keys.

> So while the Bosanquet keyboard is akin to, say,
> Esperanto, the notion of extending the Halberstadt layout
> is more like Interlingua: it is not designed to be
> "tuning neutral" or anything like that, but rather to
> pick up on and emphasize the cues that existing keyboard
> players have.

Should we all look up "Interlingua" to understand this?

> The problem is that once you get much past 12, the
> Halberstadt layout quickly becomes impractical if only
> two tiers are involved. 19-equal, for example, lends
> itself to a 12+7 layout for prioritizing meantone. This
> makes it pretty difficult to hit an octave. The solution,
> as I see it, is to come up with a three-tiered
> Halberstadt system.

Yes. And to go into the details you'll need to make a load
of prototypes to test out.

> Alternate 12-equal that has 5 as the white keys and 7 as
> the black keys. The white key widths are distorted and
> severely out of proportion because I hate ASCII art:

I firmly believe that 5 nominals would make more sense for
heavily chromatic music than 7, but I don't think anybody's
come up with a pentatonic notation for this purpose.
That's probably because diatonic scales tend to precede
chromatic ones in musical evolution. I don't think it's
worth working out the extended pentatonic notation because
the diatonic notations are too well established, but if
anybody out there's even crazier than me, why not give it a
try?

Graham

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> I started investigating how to generalize the Halberstadt layout
> for other tunings besides 12.
>
<snip>
> 3) It will probably be somewhat intuitive for existing keyboard
> players to make the switch, as the proprioceptive cues that they
> are accustomed to can also apply here.

Yes, that's a really obvious advantage. For me,. e.g., it would take much time to get as fluent on a completely new keyboard as I am on the standard.

> 4) They prioritize melodic relationships in such a way that certain
> generalized keyboards do not. 1 out of 31 on an Axis harmonic table
> layout, for example, is going to be much further away than whatever
> 5/4 is mapped to. This means that while Halberstadts aren't so
> great for ease of playing harmony, they may offer some advantages
> for melodic playing and soloing.

That makes sense to me, too.

>
> This is far from a complete analysis, but here are some ideas on
> how to extend the Halberstadt concept to a three-tiered system.
> Instead of going with complementary MOS's, I went with 22-tet's
> complementary pentachordal major + superpyth[7] + superpyth[5]
> combo, which combine to form the whole 22-tet set (thanks Paul
> Erlich).

Ey, that's a cool property. It indeed suggests that layouts based on Halberstadt can work sort of naturally for 22edo.

I would opt for superpyth[7] on bottom and superpyth[5] in the middle, since that gives a standard Halberstadt. (Although it has to be noted that the 2 superpyths do not combine to a superpyth[12] - else the complemente would not be a decatonic pentachord scale.)

I just tried out the followng: taking the Kotschy layout (http://www.newkeyboard.de) as a model and modifying it, I came up with:

https://share.ols.inode.at/Q4RVQCF214D36AKZ70W9KXS8J5ASXA8ODYJ5AY6B

The numbers indicate 22edo steps. The superpyth diatonic major scale (bottom tier) starts on 0; on the top tier, starting at 1, is a standard pentachord minor scale.

This layout has also the property that the bottom and the top tier (7 + 10) work as superpyth layout for 17edo.
--
Hans Straub

--- In tuning@yahoogroups.com, "hstraub64" <straub@...> wrote:
>
> --- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@> wrote:
> >
> > I started investigating how to generalize the Halberstadt layout
> > for other tunings besides 12.
> >
> <snip>
> > 3) It will probably be somewhat intuitive for existing keyboard
> > players to make the switch, as the proprioceptive cues that they
> > are accustomed to can also apply here.
>
> Yes, that's a really obvious advantage. For me,. e.g., it would take much time to get as fluent on a completely new keyboard as I am on the standard.

I've learned two new keyboards (Wilson-Bosanquet generalized & Moschino system accordion free bass). For the former (in which many of the patterns are very similar to the Halberstadt) it took me only about 50 hours to reach that point, and for the latter (which was much different from anything I had previously learned) it took about 150 hours). The only thing I had to unlearn with the Wilson-Bosanquet keyboard was to break myself of the habit of playing flats on the same keys as the sharps (which took only a few hours).

There are so many new intervals and chords to learn in a microtonal tuning that any keyboard not having transpositional invariance would impose a very steep learning curve for the player to become fluent in all keys.

--George

On Thu, Dec 2, 2010 at 10:05 AM, Graham Breed <gbreed@...> wrote:
>
> Mike Battaglia <battaglia01@...> wrote:
> > I started investigating how to generalize the Halberstadt
> > layout for other tunings besides 12. While there is still
> > obviously a case to be made for transpositional-invariant
> > keyboards, it is my opinion that
> > transpositionally-noninvariant keyboards should be
> > explored as well. I think that the Halberstadt, in
> > particular, has a number of distinct advantages that
> > aren't usually discussed round these parts:
>
> There's a survey of keyboards in the special microtonal
> edition of Computer Music Journal, from the 80s sometime.
> That uses the word "accretion" for this class of
> keyboards. You'll have to check the article to see exactly
> what term was used.

Thanks for the reference. I found it - it's called "History and Principles
of Microtonal Keyboards", by Douglas Keislar. George Secor and the Scalatron
get a shout out as well. I can't read it right now because my ISP is
crapping out on www.media.mit.edu, but here's the link:

http://www.media.mit.edu/resenv/classes/MAS960/NewReadings/keislar_history.pdf

> As well as the advantages you give, note that
> accretion-type keyboards tend to be a better fit to
> notation than generalized ones. I don't think I've seen a
> notation that isn't based around a scale of nominals, which
> would map to either the black or white keys.

I can't figure out what the precise definition of "accretion" is since I
can't read the article right now, but that's probably because all of the
notational systems I've seen are defined in terms of unequal scale
structures (often MOS). In our meantone-based naming convention, E-F is
shorter than D-E, but that doesn't bother anyone. We could come up with a
notation system in 12-equal based on two interlocking 6-tet scales, as in
the Janko keyboard, in which the note names are C D E F G A and their
corresponding sharps and flats, which would be tempered together in 12-equal
but not 18-equal. I find that type of structure to be overly mechanical and
perhaps tends better to a serialist composer.

I think that the notion of defining a keyboard around some "archetypical"
scale is so important to how we've been playing, thinking of, and writing
music that there's no need to do anything like the above. Maybe the best way
to go is pick some kind of maximally even 7-note scale so that all of the
7-note MOS's in the tuning system will be nearby.

> > So while the Bosanquet keyboard is akin to, say,
> > Esperanto, the notion of extending the Halberstadt layout
> > is more like Interlingua: it is not designed to be
> > "tuning neutral" or anything like that, but rather to
> > pick up on and emphasize the cues that existing keyboard
> > players have.
>
> Should we all look up "Interlingua" to understand this?

Interlingua is a constructed language that kind of looks like Latin. Its
goal is to be recognizable to the majority of Romance language speakers
without much or any additional training. It is also sort of recognizable to
"educated" English speakers who are used to dealing with Latin roots and can
recognize a lot of Interlingua words that way.

Interlingua is also biased towards Romance language speakers, and people who
speak, say, Arabic won't have an easy time learning it. Esperanto, on the
other hand, aims to be a "politically neutral" language that is easy for
everyone to learn and takes advantage of several "features" that don't exist
in natural languages. Don't ask me what they are, because I don't speak
Esperanto.

In short, Esperanto's goal is to be easily learnable, and Interlingua's goal
is to be highly comprehensible to existing Romance speakers.

> > The problem is that once you get much past 12, the
> > Halberstadt layout quickly becomes impractical if only
> > two tiers are involved. 19-equal, for example, lends
> > itself to a 12+7 layout for prioritizing meantone. This
> > makes it pretty difficult to hit an octave. The solution,
> > as I see it, is to come up with a three-tiered
> > Halberstadt system.
>
> Yes. And to go into the details you'll need to make a load
> of prototypes to test out.

I'm afraid that's where I get off the train.

> > Alternate 12-equal that has 5 as the white keys and 7 as
> > the black keys. The white key widths are distorted and
> > severely out of proportion because I hate ASCII art:
>
> I firmly believe that 5 nominals would make more sense for
> heavily chromatic music than 7, but I don't think anybody's
> come up with a pentatonic notation for this purpose.

Why do you think that?

> That's probably because diatonic scales tend to precede
> chromatic ones in musical evolution. I don't think it's
> worth working out the extended pentatonic notation because
> the diatonic notations are too well established, but if
> anybody out there's even crazier than me, why not give it a
> try?

Maybe that would have some application for superpyth[12], where you have 5+7
scales instead of 7+5.

Here's a diagram for how a 5+7 notation system might work. I came up with a
different layout for this, which I like better than the other ones: Whenever
you have two black keys adjacent to one another, make them pads, but if not,
just make them stubs, so that they're easier to play:

1# 2# 3# 4# 5# 1# 2# 3# 4# 5#
1b 2b 3b 4b 5b 1b 2b 3b 4b 5b
B C# D# E# F# G# A# B C# D# E# F# G# A#
Cb Db Eb F Gb Ab Bb Cb Db Eb F Gb Ab Bb
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ └┐ │ │ │ │ ┌┘ │ └┐ │ │ ┌┘ │ └┐ │ │ │ │ ┌┘ │ └┐ │ │ ┌┘ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
│ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │ │
├──┘ └┬┘ └┬┘ └──┼──┘ └┬┘ └──┼──┘ └┬┘ └┬┘ └──┼──┘ └┬┘ └──┤
│ │ │ │ │ │ │ │ │ │ │
│ 1 │ 2 │ 3 │ 4 │ 5 │ 1 │ 2 │ 3 │ 4 │ 5 │
│ C │ D │ E │ G │ A │ C │ D │ E │ G │ A │
│ │ │ │ │ │ │ │ │ │ │
└─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┴─────┘

I've given the notes numeric names from 1-5, and below it I have the
corresponding diatonic names written. So here's the circle of fifths:

1-4-2-5-3-1b-4b-2b-5b-3b

So in this system, 1# and 2b are tempered to be the same thing, but in
19-tet, they'd differ. Also notice that you end up getting the "flattened"
version of each note as you go around the circle of fifths, where as with a
7-based naming convention, you first end up hitting the sharps
(C-G-D-A-E-B-F#!!11one)

Lots of fun to be had here. I just wish there were an easy way to prototype
this stuff in real life.

-Mike

On Thu, Dec 2, 2010 at 2:19 PM, gdsecor <gdsecor@...> wrote:
>
>
> I've learned two new keyboards (Wilson-Bosanquet generalized & Moschino system accordion free bass). For the former (in which many of the patterns are very similar to the Halberstadt) it took me only about 50 hours to reach that point, and for the latter (which was much different from anything I had previously learned) it took about 150 hours). The only thing I had to unlearn with the Wilson-Bosanquet keyboard was to break myself of the habit of playing flats on the same keys as the sharps (which took only a few hours).
>
> There are so many new intervals and chords to learn in a microtonal tuning that any keyboard not having transpositional invariance would impose a very steep learning curve for the player to become fluent in all keys.

I would assume so, and I'm doubtful that these keyboards will do much
good for tunings with a lot of notes. 22-tet looks doable. I'm not
sure if 31 would be so successful. But then again, if one could
organize 31 into a clever 4-rank system, and one in which
self-similarity is generally present, then maybe so.

-Mike

> So while the Bosanquet keyboard is akin to, say,
> Esperanto, the notion of extending the Halberstadt layout
> is more like Interlingua: it is not designed to be
> "tuning neutral" or anything like that, but rather to
> pick up on and emphasize the cues that existing keyboard
> players have.

Let's not forget lojban! If anything its probably more comparable because it was built from the ground up with new grammar structures built from logic, not any culture or romantic/germanic/other language. Both interlingua and esperanto = Halberstadt and lojban = generalized keyboards :P

If only the day when microtonality is mainstream and we talk about it in lojban were closer than I think it is...

Mike Battaglia <battaglia01@...> wrote:

> Thanks for the reference. I found it - it's called
> "History and Principles of Microtonal Keyboards", by
> Douglas Keislar. George Secor and the Scalatron get a
> shout out as well. I can't read it right now because my
> ISP is crapping out on www.media.mit.edu, but here's the
> link:
>
> http://www.media.mit.edu/resenv/classes/MAS960/NewReadings/keislar_history.pdf

It's coming now, 20 minutes remaining.

> I can't figure out what the precise definition of
> "accretion" is since I can't read the article right now,
> but that's probably because all of the notational systems
> I've seen are defined in terms of unequal scale
> structures (often MOS). In our meantone-based naming
> convention, E-F is shorter than D-E, but that doesn't
> bother anyone. We could come up with a notation system in
> 12-equal based on two interlocking 6-tet scales, as in
> the Janko keyboard, in which the note names are C D E F G
> A and their corresponding sharps and flats, which would
> be tempered together in 12-equal but not 18-equal. I find
> that type of structure to be overly mechanical and
> perhaps tends better to a serialist composer.

Note: there's a group for discussing (mostly) chromatic
notations, where Douglas Keislar is active:

http://groups.google.com/group/musicnotation/

The unequal nature of the diatonic does, indeed, bother
some people. But, yes, two chains of 6 doesn't fit with
much existing music.

> I think that the notion of defining a keyboard around
> some "archetypical" scale is so important to how we've
> been playing, thinking of, and writing music that there's
> no need to do anything like the above. Maybe the best way
> to go is pick some kind of maximally even 7-note scale so
> that all of the 7-note MOS's in the tuning system will be
> nearby.

I haven't seen any notation that captures the generalized
nature of generalized keyboards, and I can't work out how
to do it in a remotely readable way.

> > > So while the Bosanquet keyboard is akin to, say,
> > > Esperanto, the notion of extending the Halberstadt
> > > layout is more like Interlingua: it is not designed
> > > to be "tuning neutral" or anything like that, but
> > > rather to pick up on and emphasize the cues that
> > > existing keyboard players have.
<snip>
> In short, Esperanto's goal is to be easily learnable, and
> Interlingua's goal is to be highly comprehensible to
> existing Romance speakers.

But the Bosanquet keeps the 7 note MOS in a row, so it
shouldn't be that hard to learn. People who have learned
it say it wasn't so difficult. There's also a fair bit of
Esparanto you can understand without studying it, if you
already know some European languages.

> > Yes. And to go into the details you'll need to make a
> > load of prototypes to test out.
>
> I'm afraid that's where I get off the train.

Then I'm afraid you're better off getting a generalized
keyboard and making do for all practical purposes.

> > I firmly believe that 5 nominals would make more sense
> > for heavily chromatic music than 7, but I don't think
> > anybody's come up with a pentatonic notation for this
> > purpose.
>
> Why do you think that?

Because you can get the full 12 note scale with a single
pair of accidentals, but only 5 nominals instead of 7. So
the music would be slightly more compressed, meaning you can
get more on a page, with no more cognitive load.

Graham

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Note: there's a group for discussing (mostly) chromatic
> notations, where Douglas Keislar is active:
>
> http://groups.google.com/group/musicnotation/

Great reference, thanks for that.

> The unequal nature of the diatonic does, indeed, bother
> some people. But, yes, two chains of 6 doesn't fit with
> much existing music.

I'm all for experimentation, but I don't see how it would offer any advantage. At least for the janko layout we get transposition-invariant fingering. Would a notation like that lead to some kind of "transposition-invariant naming scheme" or something? Or is it just some attempt to come up with a culturally unsympathetic notation system?

I say it really isn't all that hard to remember that the octatonic scale in 12-tet is C Db Eb E F# G A Bb C. The fact that there's two "E's" really doesn't screw things up that much.

> > I think that the notion of defining a keyboard around
> > some "archetypical" scale is so important to how we've
> > been playing, thinking of, and writing music that there's
> > no need to do anything like the above. Maybe the best way
> > to go is pick some kind of maximally even 7-note scale so
> > that all of the 7-note MOS's in the tuning system will be
> > nearby.
>
> I haven't seen any notation that captures the generalized
> nature of generalized keyboards, and I can't work out how
> to do it in a remotely readable way.

That would be an interesting project... not sure how to do that. Maybe it could use some system of accidentals which are graphically set up in such a way that you can "temper" them together. Like, say in 12, if you're on a generalized keyboard, and you're at C, you can approach a C# from at least two axes. Maybe you could set up some system of accidentals that, as you approach the C# from one axis, the accidental slowly changes, and as you approach it from a different axis, it also slowly changes, and "temper" them together such that they end up changing to be the same exact thing for C#. I dunno.

> > In short, Esperanto's goal is to be easily learnable, and
> > Interlingua's goal is to be highly comprehensible to
> > existing Romance speakers.
>
> But the Bosanquet keeps the 7 note MOS in a row, so it
> shouldn't be that hard to learn. People who have learned
> it say it wasn't so difficult. There's also a fair bit of
> Esparanto you can understand without studying it, if you
> already know some European languages.

I think that I'd really enjoy having a generalized keyboard. I think I wouldn't be happy unless I had something like a Scalatron or a similar layout. I've looked at the AXiS and it seems really weird (they've oriented the hexagons strangely), but I'll probably get one because it's cheap. I also think a generalized keyboard would be nice for anyone who wants to play in more than one tuning.

But here's another scenario where mapping out the Halberstadt would be useful: let's say that it's the year 2020 AD, and let's say this stuff has finally caught on by then. I think it's most likely that it'll start with people picking one equal temperament to obsess on for a while, and that'll be the catalyst for some kind of revolution whereby people rediscover everything that's everyone's developed on this list (hopefully academia will have caught on by then).

So let's say it's 19-tet or 22-tet or something that suddenly goes viral, and that gains a reputation amongst hip artsy college music students for being a hip new alternate "number of notes per octave" tuning system that's different than 12.

I think in that case it'll be useful for there to be some kind of "standardized" 19/22/whatever-tet accretion keyboard for people to start out with, since transpositional or tuning invariance won't be a big deal at that point yet.

And note how popular the 16-tet 9+7 Goldstein layout has become. Fleshing out the ergonomics (and maybe even aesthetic appeal) of a three-rank accretion keyboard would make practical higher-numbered ET's, which would provide for a few attractive "beginner" options for whatever lucky ET ends up in the spotlight. I have doubts accretion layouts would ever work for really high numbered ET's though (I think 31 is pushing it).

> > > Yes. And to go into the details you'll need to make a
> > > load of prototypes to test out.
> >
> > I'm afraid that's where I get off the train.
>
> Then I'm afraid you're better off getting a generalized
> keyboard and making do for all practical purposes.

If I had any idea how to build a prototype, I would. Where do I even start? Do I use legos? I don't even know where to start...

> > > I firmly believe that 5 nominals would make more sense
> > > for heavily chromatic music than 7, but I don't think
> > > anybody's come up with a pentatonic notation for this
> > > purpose.
> >
> > Why do you think that?
>
> Because you can get the full 12 note scale with a single
> pair of accidentals, but only 5 nominals instead of 7. So
> the music would be slightly more compressed, meaning you can
> get more on a page, with no more cognitive load.

Well I've posted the PDF up, so maybe you can take the concept somewhere. You'd need a much larger staff though, I think. I gave the notes names 1-5, but I think that's cliche and played out.

-Mike

Mike wrote :

> I started investigating how to generalize the Halberstadt layout > for other
> tunings besides 12. While there is still obviously a case to be > made for
> transpositional-invariant keyboards, it is my opinion that
> transpositionally-noninvariant keyboards should be explored as well.
> .../...
> A self-similar "fractal"-based approach :
> .../...

Very interesting approach, Mike, though it seems I'm not able to receive the figures correctly, wether fixed width font or not...)
Aah ! just opened your pdf ! much better indeed.
More precisely about self-similar patterns, and not only with patterns but with interval proportions, these can be found in the meta-temperament versions I have been designing for several temperaments such as Miracle (discussed on this list in february 2010), or Meantone, Superpyth, Pajara, Mohajira, Magic, Hanson, etc.
Here is for example a "fractal waveform" interpretation of Miracle temperament (in my Tuning List files folder) :
http://f1.grp.yahoofs.com/v1/sM34TGfTB_d0laIabKu82T3H_1j2lJt948CvYzCaF-5gCsU4q8fscZ930dYLAKI_7ARUYOa-43BtjXeULdkPkUxlWYWKtUB1TQ/JacquesDudon/Meta-Miracle_1258.jpg
where you can recognize several famous subsets such as Blackjack, Canasta and StudLoco as well as Mohajira, that could perhaps serve some keyboard designs. This disk was not intended to design a keyboard but to test the temperament MOS as a palette of waveforms.

I confirm that the Halberstadt layout can be modeled according to at least two different fractal algorithms, as we can see on two other disks,
where again one circle represents an octave, so you figurate the keyboards it suggests by unrolling them on a line.
These layout have to be visualised at the last row towards the instrument, were black and white keys alternate. The white keys join near the middle of the blak keys, but are not represented here except between consecutive white keys.

Let 's start with the Golden meantone disk, based on the fractal Phi waveform :
http://f1.grp.yahoofs.com/v1/sM34TAgaJc50laIaj7CsUDxWvVEhS5V1oDxir1ZFsahkrQUkYWtCNWpkVCNYDc56Ed4-RohAhhpm2bSsWETY2bWgN18SdEsR9w/JacquesDudon/Golden_1267.pdf
The 12 keys of the Halberstadt with its 5 black keys can be easily seen at the first ring from the center. But that's only for this ring, because the next developments to 19, 31, 50 tones show different patterns for the black keys, packed by two or one only, instead of packs of 2 and 3. The substitution rule between one ring and the next goes forever like this : one black interval changes into a white, and a white divides into one white and one black. This follows endlessly the Fibonacci infinite word, so this design certainly shows numerous common proportions between intervals, and is approached within an infinity of MOS levels (5, 7, 12, 19, 31, 50, 81, 131, 212..., without never been swallowed by any), but in the end it is not typically characteristic of the Halberstadt layout.

Let see the Aksaka disk now, that shows also a similar Halberstadt layout on ring 3 and its further fractal developments in 29 tones and 70 tones per octave :
http://f1.grp.yahoofs.com/v1/sM34TBjIBoN0laIatx9FhuCJ7toWrFv0symA_7s1M4kCuc6-n6oCk-2SJDoLKZWt5KkvUshNwkgQiDlYyZmlS-wFHWQ8Bi7BKA/JacquesDudon/Aksaka_1266.pdf
Now we see the Halberstadt original patterns are respected all the way and with the same proportions :
White key/Back key = minor third/wholetone = fifth/fourth = octave+fourth/octave etc. = sqrt of 2, or :
(Black key) : Whole tone : Fourth : Octave : Fourth and 2 octaves... etc. follow the same proportion of sqrt of 2 + 1.
For example, the 29 tones octave layout in its [tetrachord+wholetone+tetrachord] division uses the same design as the [octave+fourth+octave] (=12+5+12keys) of the precedent 12 tones Halberstadt layout, and the substitution rule for Aksaka's infinite word goes forever like this : one black interval divides into a white+black, and a white divides into two whites and one black.
Another valid sequence, such as [F#... C] could also follow the same patterns with respectively 7, 17, 41, or 99 keys.
While with the Golden meantone, the numbers of white and black keys were converging towards a Phi proportion, in Aksaka sequences the numbers of white keys over the black keys converges to sqrt of 2 ~1,414.
Of course past 12 notes the octaves would be quite unplayable I admit, unless for guys with very large hands, but as you very well showed in your propositions certainly three rows could be arranged here too.
It must be mentionned that these fractal layouts do not only define keys arrangements, but mainly fractal proportions between intervals and therefore suggest very precise generator solutions, for what I call "meta-versions" of the temperaments. Those are of course not tuning obligations for a keyboard but interesting to know as central temperaments :
The Golden meantone meta-temperament fifth generator is 696.214473955 c.,
while Aksaka's slightly extended fifth is worth of (2 - sqrtof 2) octaves or 702.9437253 c., which qualifies best, according to Graham's list, for Undecental or perhaps even Dominant temperaments. That's funny, because the Halberstadt keyboard served mostly for meantone, and as we see the more specific fractal model of Halberstadt is not a Meantone. We can verify the main MOS cycles of this generator are 12, 29, 70, with secondary cycles at 17, 41, 99.

Both of these fractal patterns I have been using for sounds and rhythms since the beginning of my photosonic disks and both are excellent.
I would like to say more about the rhythms and why I call the second one "Akaska", but I must go, it will be for a another time.
- - - - - - - -
Jacques

"battaglia01" <battaglia01@...> wrote:
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...>
> wrote:
>
> > The unequal nature of the diatonic does, indeed, bother
> > some people. But, yes, two chains of 6 doesn't fit with
> > much existing music.
>
> I'm all for experimentation, but I don't see how it would
> offer any advantage. At least for the janko layout we get
> transposition-invariant fingering. Would a notation like
> that lead to some kind of "transposition-invariant naming
> scheme" or something? Or is it just some attempt to come
> up with a culturally unsympathetic notation system?

The double-6 notation was an Aunt Sally you set up to knock
down. Now your talking about it as a serious idea from
somebody else.

Chromatic notations either have 12 nominals on the staff or
a single line with something like numbers for the notes.
None of them have taken off unless you count guitar
tablature. But they are seriously proposed and the do have
advantages if your music is heavily 12-based.

> I say it really isn't all that hard to remember that the
> octatonic scale in 12-tet is C Db Eb E F# G A Bb C. The
> fact that there's two "E's" really doesn't screw things
> up that much.

No, but it's another thing to clutter your mind with.
It's easier to remember it as a 1 2 1 2 1 2 ... pattern on
a key or fretboard.

> > I haven't seen any notation that captures the
> > generalized nature of generalized keyboards, and I
> > can't work out how to do it in a remotely readable way.
>
> That would be an interesting project... not sure how to
> do that. Maybe it could use some system of accidentals
> which are graphically set up in such a way that you can
> "temper" them together. Like, say in 12, if you're on a
> generalized keyboard, and you're at C, you can approach a
> C# from at least two axes. Maybe you could set up some
> system of accidentals that, as you approach the C# from
> one axis, the accidental slowly changes, and as you
> approach it from a different axis, it also slowly
> changes, and "temper" them together such that they end up
> changing to be the same exact thing for C#. I dunno.

I don't know how what you suggest would work, which is
really the point. You could also use a pair of numbers to
specify each interval. But who can read that at a
keyboard? It's easier to stick with conventional notation.

Graham

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> I would like to say more about the rhythms and why I call the second
> one "Akaska", but I must go, it will be for a another time.
> - - - - - - - -
> Jacques

I mean "AKSAKA", in the non-reversed version.
If you play the strong beats on the blacks on the Halberstadt ring (counter-clockwise will be in better phase with the other rings) you get a succession of two Aksak rhythms (from the Balkans) of 7 and 5 beats, forming a typical and wonderful Aka (= Pygmy ethnic group) rhythm of 12 beats.

http://f1.grp.yahoofs.com/v1/0On4TLanS_8zGxrn-86bhCZeG7x-UFRSK3zwv_wxuPWvQKyoMvN3olPhNWQj_7YzW0FDlPBuLhBJsB3OYVwtzShGcXcD7OpcGw/JacquesDudon/Aksaka_1266.pdf

--- In tuning@yahoogroups.com, "battaglia01" <battaglia01@...> wrote:

> > The unequal nature of the diatonic does, indeed, bother
> > some people. But, yes, two chains of 6 doesn't fit with
> > much existing music.
>
> I'm all for experimentation, but I don't see how it would offer any advantage.

The fact that there are only two fingers for each scale--the one with the tonic on a white key, and the one with the tonic on a black key--strikes me as an advantage.

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> Here is for example a "fractal waveform" interpretation of Miracle
> temperament (in my Tuning List files folder) :
> http://f1.grp.yahoofs.com/v1/
> sM34TGfTB_d0laIabKu82T3H_1j2lJt948CvYzCaF-5gCsU4q8fscZ930dYLAKI_7ARUYOa-
> 43BtjXeULdkPkUxlWYWKtUB1TQ/JacquesDudon/Meta-Miracle_1258.jpg

Your links seem to be broken, probably because the urls are too long. I suggest using tinyurl.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Thu, Dec 2, 2010 at 2:19 PM, gdsecor <gdsecor@...> wrote:
> >
> >
> > I've learned two new keyboards (Wilson-Bosanquet generalized & Moschino system accordion free bass). For the former (in which many of the patterns are very similar to the Halberstadt) it took me only about 50 hours to reach that point, and for the latter (which was much different from anything I had previously learned) it took about 150 hours). The only thing I had to unlearn with the Wilson-Bosanquet keyboard was to break myself of the habit of playing flats on the same keys as the sharps (which took only a few hours).
> >
> > There are so many new intervals and chords to learn in a microtonal tuning that any keyboard not having transpositional invariance would impose a very steep learning curve for the player to become fluent in all keys.
>
> I would assume so, and I'm doubtful that these keyboards will do much
> good for tunings with a lot of notes. 22-tet looks doable. I'm not
> sure if 31 would be so successful. But then again, if one could
> organize 31 into a clever 4-rank system, and one in which
> self-similarity is generally present, then maybe so.

31 is much easier than 22 at the 5 limit, and the two are about equally easy at both the 7 and 11 limits. For a tempered 4:5:6:7:9:11 chord, 31 actually spans fewer generators in the chain of fifths than does 22, so the amount of vertical (y-axis) reach is less; however, whenever alternate (duplicate) keys are used, the duplicates are one row closer in 22. All things considered, 31 and 22 are about the same difficulty.

OTOH, 29 and schismatic temperaments are more difficult than 31, due to the relative locations of the primes. (I don't have much interest in 29-EDO, but I find that it's a nice division for mapping JI and near-JI sets of tones.)

17, of course, is as easy as pie (and also a nice alternative for mapping JI).

For anything above 31 tones/octave, I would recommend using a different keyboard, because the duplicate keys get much farther apart.

--George

On Thu, Dec 2, 2010 at 10:56 AM, hstraub64 <straub@...> wrote:
>
> I just tried out the followng: taking the Kotschy layout (http://www.newkeyboard.de) as a model and modifying it, I came up with:
>
> https://share.ols.inode.at/Q4RVQCF214D36AKZ70W9KXS8J5ASXA8ODYJ5AY6B
>
> The numbers indicate 22edo steps. The superpyth diatonic major scale (bottom tier) starts on 0; on the top tier, starting at 1, is a standard pentachord minor scale.

Hey, this is really cool. I've never seen this before. Although I'd
say that I still would probably find 9+7 layouts easier for 16-equal
than 4+4+8 layouts. Good find - has this been posted before?

-Mike

On Fri, Dec 3, 2010 at 6:48 AM, Jacques Dudon <fotosonix@...> wrote:
>
> More precisely about self-similar patterns, and not only with patterns but with interval proportions, these can be found in the meta-temperament versions I have been designing for several temperaments such as Miracle (discussed on this list in february 2010), or Meantone, Superpyth, Pajara, Mohajira, Magic, Hanson, etc.
> Here is for example a "fractal waveform" interpretation of Miracle temperament (in my Tuning List files folder) :
> http://f1.grp.yahoofs.com/v1/sM34TGfTB_d0laIabKu82T3H_1j2lJt948CvYzCaF-5gCsU4q8fscZ930dYLAKI_7ARUYOa-43BtjXeULdkPkUxlWYWKtUB1TQ/JacquesDudon/Meta-Miracle_1258.jpg

The link got screwed up - it's supposed to be this:
/tuning/files/JacquesDudon/Meta-Miracle_1258.jpg

> where you can recognize several famous subsets such as Blackjack, Canasta and  StudLoco as well as Mohajira, that could perhaps serve some keyboard designs. This disk was not intended to design a keyboard but to test the temperament MOS as a palette of waveforms.
> I confirm that the Halberstadt layout can be modeled according to at least two different fractal algorithms, as we can see on two other disks,
> where again one circle represents an octave, so you figurate the keyboards it suggests by unrolling them on a line.

Man, this is fascinating. So what is this, "meta-miracle"? Is this
like a "golden Miracle" or something? But this is exactly what I was
talking about - coming up with a fractalized representation of ALL of
the MOS's of a tuning, and then picking a subset of that to represent.
The only difference is that sometimes you'd have to represent some
MOS's twice (like 17-tet is superpyth[7] + superpyth[5] + superpyth[5]
--- superpyth[10] isn't an MOS).

How did you generate this?

> These layout have to be visualised at the last row towards the instrument, were black and white keys alternate. The white keys join near the middle of the blak keys, but are not represented here except between consecutive white keys.
> Let 's start with the Golden meantone disk, based on the fractal Phi waveform :
> http://f1.grp.yahoofs.com/v1/sM34TAgaJc50laIaj7CsUDxWvVEhS5V1oDxir1ZFsahkrQUkYWtCNWpkVCNYDc56Ed4-RohAhhpm2bSsWETY2bWgN18SdEsR9w/JacquesDudon/Golden_1267.pdf

Fixed link: /tuning/files/JacquesDudon/Golden_1267.pdf

So it represents the white keys of 19 as 5+2+3+2, and the black as 4+3 then...

> The 12 keys of the Halberstadt with its 5 black keys can be easily seen at the first ring from the center. But that's only for this ring, because the next developments to 19, 31, 50 tones show different patterns for the black keys, packed by two or one only, instead of packs of 2 and 3. The substitution rule between one ring and the next goes forever like this : one black interval changes into a white, and a white divides into one white and one black. This follows endlessly the Fibonacci infinite word, so this design certainly shows numerous common proportions between intervals, and is approached within an infinity of MOS levels (5, 7, 12, 19, 31, 50, 81, 131, 212..., without never been swallowed by any), but in the end it is not typically characteristic of the Halberstadt layout.

But perhaps if you flipped it around - so that 7 were on the outside,
and it increased infinitely towards the center instead of outward,
then perhaps it would look more like a Halberstadt. And also if you
unwrapped it, so that it was linear instead of on a disc... The inner
ring looks like a Halberstadt to me, it's got ,',',,',',', in it.

Also, how might you modify this to work for a three-tiered layout...?

> Let see the Aksaka disk now, that shows also a similar Halberstadt layout on ring 3 and its further fractal developments in 29 tones and 70 tones per octave :
> http://f1.grp.yahoofs.com/v1/sM34TBjIBoN0laIatx9FhuCJ7toWrFv0symA_7s1M4kCuc6-n6oCk-2SJDoLKZWt5KkvUshNwkgQiDlYyZmlS-wFHWQ8Bi7BKA/JacquesDudon/Aksaka_1266.pdf

Fixed link: /tuning/files/JacquesDudon/Aksaka_1266.pdf

This definitely does look more like a Halberstadt, but I think because
it tends to cluster the keys in terms of 3 white/2 black and 4 white/3
black, whereas the Golden Meantone one would cluster them in terms of
2 white/1 black a lot of the time.

But this is really fascinating, can you explain exactly what you're
doing and how you generated this diagram? How do you define what's
white and what's black?

> Now we see the Halberstadt original patterns are respected all the way and with the same proportions :
> White key/Back key = minor third/wholetone = fifth/fourth = octave+fourth/octave etc. = sqrt of 2,  or :
> (Black key) : Whole tone : Fourth : Octave : Fourth and 2 octaves... etc. follow the same proportion of sqrt of 2 + 1.
> For example, the 29 tones octave layout in its [tetrachord+wholetone+tetrachord] division uses the same design as the [octave+fourth+octave] (=12+5+12keys) of the precedent 12 tones Halberstadt layout, and the substitution rule for Aksaka's infinite word goes forever like this : one black interval divides into a white+black, and a white divides into two whites and one black.

I see, but the layout doesn't seem to "line up" as with the Golden
Meantone - is this an alignment issue, or is that how it's supposed to
look.

> Another valid sequence, such as [F#... C] could also follow the same patterns with respectively 7, 17, 41, or 99 keys.
> While with the Golden meantone, the numbers of white and black keys were converging towards a Phi proportion, in Aksaka sequences the numbers of white keys over the black keys converges to sqrt of 2 ~1,414.
> Of course past 12 notes the octaves would be quite unplayable I admit, unless for guys with very large hands, but as you very well showed in your propositions certainly three rows could be arranged here too.

How did you come up with this Aksaka layout? How did you know it would
work here?

> It must be mentionned that these fractal layouts do not only define keys arrangements, but mainly fractal proportions between intervals and therefore suggest very precise generator solutions, for what I call "meta-versions" of the temperaments. Those are of course not tuning obligations for a  keyboard but interesting to know as central temperaments  :
> The Golden meantone meta-temperament fifth generator is 696.214473955 c.,
> while Aksaka's slightly extended fifth is worth of (2 - sqrtof 2) octaves or 702.9437253 c., which qualifies best, according to Graham's list, for Undecental or perhaps even Dominant temperaments. That's funny, because the Halberstadt keyboard served mostly for meantone, and as we see the more specific fractal model of Halberstadt is not a Meantone. We can verify the main MOS cycles of this generator are 12, 29, 70, with secondary cycles at 17, 41, 99.

When you say "the more specific fractal model," you mean that it
produces combinations of ,',', and ,',',',? And what commas does
undecental temper out...?

> Both of these fractal patterns I have been using for sounds and rhythms since the beginning of my photosonic disks and both are excellent.
> I would like to say more about the rhythms and why I call the second one "Akaska", but I must go, it will be for a another time.

Alright, I would like to hear more about this.

-Mike

On Fri, Dec 3, 2010 at 7:14 AM, Graham Breed <gbreed@...> wrote:
> > I say it really isn't all that hard to remember that the
> > octatonic scale in 12-tet is C Db Eb E F# G A Bb C. The
> > fact that there's two "E's" really doesn't screw things
> > up that much.
>
> No, but it's another thing to clutter your mind with.
> It's easier to remember it as a 1 2 1 2 1 2 ... pattern on
> a key or fretboard.

Not if you're using it over something like a dom7 chord... Then it's
really helpful to think of it in terms of a dom7 chord with a b9/#9,
#11, nat13. And also how it relates to other close, related meantone
modal scales, like say the altered scale or lydian dominant.

-Mike

On Fri, Dec 3, 2010 at 12:48 PM, genewardsmith
<genewardsmith@...> wrote:
>
> The fact that there are only two fingers for each scale--the one with the tonic on a white key, and the one with the tonic on a black key--strikes me as an advantage.

I meant in the sense of having some kind of notation system based
around double-6. Notation right now is based around meantone[7], and I
thought setting it up around two sets of 6-et (if you're in 12-et)
might lead to some kind of transpositionally-invariant notation, but I
don't see how.

-Mike

A transpositionally-invariant notation? I can't even conceive of what that would look like.

-Igs

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Dec 3, 2010 at 12:48 PM, genewardsmith
> <genewardsmith@...> wrote:
> >
> > The fact that there are only two fingers for each scale--the one with the tonic on a white key, and the one with the tonic on a black key--strikes me as an advantage.
>
> I meant in the sense of having some kind of notation system based
> around double-6. Notation right now is based around meantone[7], and I
> thought setting it up around two sets of 6-et (if you're in 12-et)
> might lead to some kind of transpositionally-invariant notation, but I
> don't see how.
>
> -Mike
>

> A transpositionally-invariant notation? I can't even conceive
> of what that would look like.
>
> -Igs

In 12-ET, if you have nominals assigned to whole tones
(A B C D E F) and one accidental assigned to the half-step,
that would do it.

Key signatures also kindof do the job.

At least, that's my interpretation. Maybe others have
different ideas.

-Carl

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > A transpositionally-invariant notation? I can't even conceive
> > of what that would look like.
> >
> > -Igs
>
> In 12-ET, if you have nominals assigned to whole tones
> (A B C D E F) and one accidental assigned to the half-step,
> that would do it.

It would do something, at any rate. But why stop there? What about four nominals, plus flats and sharps?

On Fri, Dec 3, 2010 at 2:12 PM, gdsecor <gdsecor@...> wrote:
> >
> > I would assume so, and I'm doubtful that these keyboards will do much
> > good for tunings with a lot of notes. 22-tet looks doable. I'm not
> > sure if 31 would be so successful. But then again, if one could
> > organize 31 into a clever 4-rank system, and one in which
> > self-similarity is generally present, then maybe so.
>
> 31 is much easier than 22 at the 5 limit, and the two are about equally easy at both the 7 and 11 limits. For a tempered 4:5:6:7:9:11 chord, 31 actually spans fewer generators in the chain of fifths than does 22, so the amount of vertical (y-axis) reach is less; however, whenever alternate (duplicate) keys are used, the duplicates are one row closer in 22. All things considered, 31 and 22 are about the same difficulty.

Are you talking about with a Bosanquet layout, or with a
Halberstadt...? I meant that I think setting up an "accretion" design
might work well for certain tunings, say 19 or 22, just like the 9+7
layout Halberstadt design is quickly becoming popular for 16-tet. I
have doubts a Halberstadt design would be useful for something like
53-tet (you'd have to have like what, 5-6 ranks of keys to make that
work? That's a pretty steep learning curve). A generalized keyboard
might be better there.

-Mike

On Sat, Dec 4, 2010 at 3:02 AM, genewardsmith
<genewardsmith@...> wrote:
>
> It would do something, at any rate. But why stop there? What about four nominals, plus flats and sharps?

So you'd have C Eb F# A as 1 2 3 4, and C major would be 1 2# 3#.
Ebmajor would be 2 3# 4#. F# major would be 3 4# 1#.

C# major would be 1# 2x 3x.

I guess it really is transpositionally invariant. An interesting idea
indeed. Is there any way to generalize something like this for prime
number ET's?

Mike wrote :

> On Fri, Dec 3, 2010 at 6:48 AM, Jacques Dudon > <fotosonix@...> wrote:
> >
> > More precisely about self-similar patterns, and not only with > patterns but with interval proportions, these can be found in the > meta-temperament versions I have been designing for several > temperaments such as Miracle (discussed on this list in february > 2010), or Meantone, Superpyth, Pajara, Mohajira, Magic, Hanson, etc.
> > Here is for example a "fractal waveform" interpretation of > Miracle temperament (in my Tuning List files folder) :
> > http://f1.grp.yahoofs.com/v1/> sM34TGfTB_d0laIabKu82T3H_1j2lJt948CvYzCaF-5gCsU4q8fscZ930dYLAKI_7ARUYO> a-43BtjXeULdkPkUxlWYWKtUB1TQ/JacquesDudon/Meta-Miracle_1258.jpg
>
> The link got screwed up - it's supposed to be this:
> /tuning/files/JacquesDudon/Meta-> Miracle_1258.jpg

How do you make this work ? I couldn't find other ways and when I tested it, it opened the file :-(

> > where you can recognize several famous subsets such as Blackjack, > Canasta and StudLoco as well as Mohajira, that could perhaps serve > some keyboard designs. This disk was not intended to design a > keyboard but to test the temperament MOS as a palette of waveforms.
> > I confirm that the Halberstadt layout can be modeled according to > at least two different fractal algorithms, as we can see on two > other disks,
> > where again one circle represents an octave, so you figurate the > keyboards it suggests by unrolling them on a line.
>
> Man, this is fascinating. So what is this, "meta-miracle"? Is this
> like a "golden Miracle" or something?

Same principle, except not based on Phi but on another fractal I found much more pertinent with Miracle.
The same substitutions and patterns of division will appear endlessly at different interval scales, this is why we can say it's a fractal.
It was explained in detail on this list :
/tuning/topicId_85954.html#86310

> But this is exactly what I was
> talking about - coming up with a fractalized representation of ALL of
> the MOS's of a tuning, and then picking a subset of that to represent.
> The only difference is that sometimes you'd have to represent some
> MOS's twice (like 17-tet is superpyth[7] + superpyth[5] + superpyth[5]
> --- superpyth[10] isn't an MOS).
>
> How did you generate this?

With my photosonic disk programm, simply by designing the right fractal waveform seed and applying the right frequencies of repetitions.
I may also overlap some of the rings in order to have smoother transitions and add basses and timber complexity here and there.

> > These layout have to be visualised at the last row towards the > instrument, were black and white keys alternate. The white keys > join near the middle of the blak keys, but are not represented here > except between consecutive white keys.
> > Let 's start with the Golden meantone disk, based on the fractal > Phi waveform :
> > http://f1.grp.yahoofs.com/v1/> sM34TAgaJc50laIaj7CsUDxWvVEhS5V1oDxir1ZFsahkrQUkYWtCNWpkVCNYDc56Ed4-> RohAhhpm2bSsWETY2bWgN18SdEsR9w/JacquesDudon/Golden_1267.pdf
>
> Fixed link: /tuning/files/> JacquesDudon/Golden_1267.pdf
>
> So it represents the white keys of 19 as 5+2+3+2, and the black as 4> +3 then...
>
> > The 12 keys of the Halberstadt with its 5 black keys can be > easily seen at the first ring from the center. But that's only for > this ring, because the next developments to 19, 31, 50 tones show > different patterns for the black keys, packed by two or one only, > instead of packs of 2 and 3. The substitution rule between one ring > and the next goes forever like this : one black interval changes > into a white, and a white divides into one white and one black. > This follows endlessly the Fibonacci infinite word, so this design > certainly shows numerous common proportions between intervals, and > is approached within an infinity of MOS levels (5, 7, 12, 19, 31, > 50, 81, 131, 212..., without never been swallowed by any), but in > the end it is not typically characteristic of the Halberstadt layout.
>
> But perhaps if you flipped it around - so that 7 were on the outside,
> and it increased infinitely towards the center instead of outward,
> then perhaps it would look more like a Halberstadt. And also if you
> unwrapped it, so that it was linear instead of on a disc... The inner
> ring looks like a Halberstadt to me, it's got ,',',,',',', in it.
>
> Also, how might you modify this to work for a three-tiered layout...?

If you consider two successive rings and place the new blacks on a third tier, it seems to be working at all levels.
The 19-keyboard for examle would have a regular 7+5 keys Halberstadt on the first two tiers, and a third (diatonic here) tier of 7 keys.
It would even allow its own new 5 black keys on a fourth tier to make a 24 tones / octave keyboard.

> > Let see the Aksaka disk now, that shows also a similar > Halberstadt layout on ring 3 and its further fractal developments > in 29 tones and 70 tones per octave :
> > http://f1.grp.yahoofs.com/v1/> sM34TBjIBoN0laIatx9FhuCJ7toWrFv0symA_7s1M4kCuc6-> n6oCk-2SJDoLKZWt5KkvUshNwkgQiDlYyZmlS-wFHWQ8Bi7BKA/JacquesDudon/> Aksaka_1266.pdf
>
> Fixed link: /tuning/files/> JacquesDudon/Aksaka_1266.pdf
>
> This definitely does look more like a Halberstadt, but I think because
> it tends to cluster the keys in terms of 3 white/2 black and 4 white/3
> black, whereas the Golden Meantone one would cluster them in terms of
> 2 white/1 black a lot of the time.

Exactly.

> But this is really fascinating, can you explain exactly what you're
> doing and how you generated this diagram? How do you define what's
> white and what's black?

The first ring (in the center) defines the colors proportions. All others come from repetitions of this seed by the same fractal ratio.

> > Now we see the Halberstadt original patterns are respected all > the way and with the same proportions :
> > White key/Back key = minor third/wholetone = fifth/fourth = octave> +fourth/octave etc. = sqrt of 2, or :
> > (Black key) : Whole tone : Fourth : Octave : Fourth and 2 > octaves... etc. follow the same proportion of sqrt of 2 + 1.
> > For example, the 29 tones octave layout in its [tetrachord> +wholetone+tetrachord] division uses the same design as the [octave> +fourth+octave] (=12+5+12keys) of the precedent 12 tones > Halberstadt layout, and the substitution rule for Aksaka's infinite > word goes forever like this : one black interval divides into a > white+black, and a white divides into two whites and one black.
>
> I see, but the layout doesn't seem to "line up" as with the Golden
> Meantone - is this an alignment issue, or is that how it's supposed to
> look.

I choosed this precise seed to line up on the center of the black keys. There are other possibilities, some would need re-phasing, and this one was faster !

> > Another valid sequence, such as [F#... C] could also follow the > same patterns with respectively 7, 17, 41, or 99 keys.
> > While with the Golden meantone, the numbers of white and black > keys were converging towards a Phi proportion, in Aksaka sequences > the numbers of white keys over the black keys converges to sqrt of > 2 ~1,414.
> > Of course past 12 notes the octaves would be quite unplayable I > admit, unless for guys with very large hands, but as you very well > showed in your propositions certainly three rows could be arranged > here too.
>
> How did you come up with this Aksaka layout? How did you know it would
> work here?

It's as I said, this is one of the first fractal waveforms I found, I've been carving optical sequencer disks with it and I couldn't miss its Aksak-Aka rhythms. And the Halberstadt pattern is basically an Aka rhythm...

> > It must be mentionned that these fractal layouts do not only > define keys arrangements, but mainly fractal proportions between > intervals and therefore suggest very precise generator solutions, > for what I call "meta-versions" of the temperaments. Those are of > course not tuning obligations for a keyboard but interesting to > know as central temperaments :
> > The Golden meantone meta-temperament fifth generator is > 696.214473955 c.,
> > while Aksaka's slightly extended fifth is worth of (2 - sqrtof 2) > octaves or 702.9437253 c., which qualifies best, according to > Graham's list, for Undecental or perhaps even Dominant > temperaments. That's funny, because the Halberstadt keyboard served > mostly for meantone, and as we see the more specific fractal model > of Halberstadt is not a Meantone. We can verify the main MOS cycles > of this generator are 12, 29, 70, with secondary cycles at 17, 41, 99.
>
> When you say "the more specific fractal model," you mean that it
> produces combinations of ,',', and ,',',',?

Yes, the full Halberstadt pattern gets repeated at all scales as part of the Akasaka infinite word, while with The Golden scale it was only an "accident". But of course as you know there are no accidents...

> And what commas does
> undecental temper out...?

Good question, but I rather leave it to Gene, Graham, Herman or other specialists, as well as this subsidiary question :
What are the differences with Kwai, Superpyth, Quasisuper, etc., (and Dominant, that I found is given with two very different generator values such as 496.746 and 498.427 c.) ?
What I know is that this "Aksaka meta-temperament" generator has to be precisely (sqrtof2 - 1) octave = 497.056274848 c., with best MOS at 5, 12, 29, 70, 169, 408... and if some of those could apply to precise linear temperaments, I am curious.

> > Both of these fractal patterns I have been using for sounds and > rhythms since the beginning of my photosonic disks and both are > excellent.
> > I would like to say more about the rhythms and why I call the > second one "Akaska", but I must go, it will be for a another time.
>
> Alright, I would like to hear more about this.
>
> -Mike

Replied already - musical excerpts to come also some time !
- - - - - - -
Jacques

--- In tuning@yahoogroups.com, Jacques Dudon <fotosonix@...> wrote:

> > And what commas does
> > undecental temper out...?

5120/5103 and 235298/234375. Not the greatest temperament in the world, but insisting on fifths as a generator limits your selection.

> What I know is that this "Aksaka meta-temperament" generator has to
> be precisely (sqrtof2 - 1) octave = 497.056274848 c., with best MOS
> at 5, 12, 29, 70, 169, 408... and if some of those could apply to
> precise linear temperaments, I am curious.

(2-sqrt(2))/2 is a generator for hemififths temperament, if that is of any use.

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:

> (2-sqrt(2))/2 is a generator for hemififths temperament, if that is of any use.

This is exactly what I found 2 days ago, and it is totally useful for a "Hemififth meta-temperament" ! :
1200 / (sqrt(2) + 2) = 351.472 cents
(Hemififth optimal generator with pure octaves = 351.477 cents...)
In other words, sqrt(2) + 2 = 3.414213562 of these Hemififth generators makes an octave.
A magic number because the solution of a major fractal algorithm : x^2 = 4x - 2, or "Persi" (because following the Parsley leafs progression)
It has a P sequence and its own infinite word following the rule :
a ->aaba, b->ab, allowing for endless self-similar patterns.
MOS cycles are 3, 7, 10, 17, 24, 41, 58, 99, 140, 239...
(more or less the same ones as Aksaka-meta temp)
Both are completely related, and same octave divisions (and same keyboards obviously) would do for both as long as the fifth is expressed with a even number of steps. The two generators proportion is sqrt(2), such as with :
7/5, 10/7, 17/12, 24/17, 41/29, 58/41, 99/70 etc. steps...
- - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - - -
Jacques

--- In tuning@yahoogroups.com, "Jacques Dudon" <fotosonix@...> wrote:

> > (2-sqrt(2))/2 is a generator for hemififths temperament, if that is of any use.
>
> This is exactly what I found 2 days ago, and it is totally useful for a "Hemififth meta-temperament" ! :
> 1200 / (sqrt(2) + 2) = 351.472 cents
> (Hemififth optimal generator with pure octaves = 351.477 cents...)

Presumably you mean pure-octaves TE tuning. The 7-limit least squares tuning is also very close at 351.471 cents.

> In other words, sqrt(2) + 2 = 3.414213562 of these Hemififth generators makes an octave.

It might also be worth noting that 1 - (2-sqrt(2))/2 = sqrt(2)/2.

> A magic number because the solution of a major fractal algorithm : x^2 = 4x - 2, or "Persi" (because following the Parsley leafs progression)
> It has a P sequence and its own infinite word following the rule :
> a ->aaba, b->ab, allowing for endless self-similar patterns.
> MOS cycles are 3, 7, 10, 17, 24, 41, 58, 99, 140, 239...
> (more or less the same ones as Aksaka-meta temp)

And exactly the ones of hemififths.

--- In tuning@yahoogroups.com, Mike Battaglia <battaglia01@...> wrote:
>
> On Fri, Dec 3, 2010 at 2:12 PM, gdsecor <gdsecor@...> wrote:
> > >
> > > I would assume so, and I'm doubtful that these keyboards will do much
> > > good for tunings with a lot of notes. 22-tet looks doable. I'm not
> > > sure if 31 would be so successful. But then again, if one could
> > > organize 31 into a clever 4-rank system, and one in which
> > > self-similarity is generally present, then maybe so.
> >
> > 31 is much easier than 22 at the 5 limit, and the two are about equally easy at both the 7 and 11 limits. For a tempered 4:5:6:7:9:11 chord, 31 actually spans fewer generators in the chain of fifths than does 22, so the amount of vertical (y-axis) reach is less; however, whenever alternate (duplicate) keys are used, the duplicates are one row closer in 22. All things considered, 31 and 22 are about the same difficulty.
>
> Are you talking about with a Bosanquet layout, or with a
> Halberstadt...?

I'm talking about a Bosanquet layout.

> I meant that I think setting up an "accretion" design
> might work well for certain tunings, say 19 or 22, just like the 9+7
> layout Halberstadt design is quickly becoming popular for 16-tet. I
> have doubts a Halberstadt design would be useful for something like
> 53-tet (you'd have to have like what, 5-6 ranks of keys to make that
> work? That's a pretty steep learning curve). A generalized keyboard
> might be better there.

I can see no advantage for an "accretion" design for any tuning whatsoever. To start with, the Halbertstadt is much less efficient than than a Bosanquet keyboard even for 12. E.g., with a Halberstadt, you have 4 different patterns for a major 3rd (C-E, D-F#, Eb-G, and F#-A#), and dissimilar intervals also share these same patterns (D-F same pattern as C-E, F-Bb same as D-F#, F#-B same as G#-C). When you add keys for something even as simple as 19, it gets even more complicated, because you have to learn even more patterns for a major 3rd (e.g., C#-E#, D#-Fx=Gb), and some patterns will have to re-learned, such as Db-F, which will no longer work with the C#-F pattern, since Db is no longer on the C# key. This also carries over to other intervals, such as the minor 3rd, major 2nd, major & minor 6th, etc. Plus, you have to learn new patterns in each of 19 keys for the more dissonant intervals in 19, such as augmented 5th, augmented 6th, augmented 2nd, and diminished 7th. IMO, it's a nightmare!

With the Bosanquet, it takes no more than a minute to learn the 5+7+5 (dark-light-dark) key pattern that clearly identifies the sharp, natural, and flat keys. Then add a row of 7 light keys above & below this to include double-sharps and double-flats. Anything you learn in one key is instantly transposable to any other key.

After you've learned 19, you can use the same keyboard for 22. The primes will be in different locations than with 19, so you'll need to learn a new set of patterns for the intervals, but you'll need to learn the patterns in only one key.

I've never experiences any confusion having different patterns for different tunings. I find that the differences in "mood" between tunings is a sufficient reminder of how to navigate the keyboard to play various intervals. For example, with 22 you're constantly aware of the exaggerated (54-cent) comma and widely tempered 5:6 (minor 3rd), which require fingering patterns unique to 22. Another cue is the very different sound of the intervals approximating ratios of 7 in 19 vs. 22 vs. 31.

--George

Hi all,

For visual entertainment and eventual microtonal applications, one simple representation of fractal Persi, alias Hemififth meta-temperament :

http://f1.grp.yahoofs.com/v1/cPr8TP85YgS2v5YlJ_rbB7qbAbykx0DA1OMuTJ5xVA8iPW52h--EnqHoQvtSM3et3HWPzSm8DHdS0nk04-7WP2YSzzjH65piog/JacquesDudon/Persi-Hemififth_1269.pdf

(Find it my TL files)

1 circle = 1 octave
1st black arc in the center = hemififth generator -
as we can see on the following rings it becomes divided like the octave is (LLLs) :
octave = 3 hemififths (~49/40) and one neutral second (~99/91) ;
hemififth = 3 semitones (~ 52/49) and one quartertone (~41/40) ;
etc. forever in same proportions.
Zooming back we could also say that the generator is one octave, inside a period (= 1 circle) of 3 octaves and a fourth.
With the beginning of black arcs the sequence describes systems of 3, 10, 3, 116 notes /octave
or 4, 14, 48, 164 in option (when using the narrower blacks).
Considering the white arcs as white keys and the black ones as black keys, it suggests also some funny keyboards with 7, 24, 82, etc. keys/octave.

This second disk is the second option I mentionned for the Aksaka_1266 previously described :

http://f1.grp.yahoofs.com/v1/cPr8TFSOwMa2v5YlULZs2tpe5oIhXJ0EvVslPwT7QmfJtEyrGvFI9nN0CRkhVk3z_lmzXv4itAjVMRiqeUP4BubGnYoIq8U_bA/JacquesDudon/Aksaka-Hemififth_1268.pdf

with a 3, 7, 17, 41, 99 (alternatively total keys, or number of black keys) sequence instead of the former 2 5 12 29 70 that were not integrating the hemififth.
It's not a Persi but an Aksaka sequence, with the interest in this one to integrate the hemififths (that's for example the interval between the most lined-up black keys), and this with the Halberstadt pattern again. Though been impractical with that number of keys, the hemififths for ex. in the 41 keys (3rd ring) would be easy to find for a standard keyboard user, as they mimic the pattern of one octave in a Halberstadt keyboard.
Pretty much fractal as well, based on a Aksaka division of hemififths.
- - - - - - - -
Jacques

Gene wrote :

> --- In tuning@yahoogroups.com, "Jacques Dudon" <fotosonix@...> wrote:
>
> > > (2-sqrt(2))/2 is a generator for hemififths temperament, if > that is of any use.
> >
> > This is exactly what I found 2 days ago, and it is totally useful > for a "Hemififth meta-temperament" ! :
> > 1200 / (sqrt(2) + 2) = 351.472 cents
> > (Hemififth optimal generator with pure octaves = 351.477 cents...)
>
> Presumably you mean pure-octaves TE tuning. The 7-limit least > squares tuning is also very close at 351.471 cents.

Difficult to find better ! - quite a rare coincidence between a temperament and such a basic fractal waveform.

> > In other words, sqrt(2) + 2 = 3.414213562 of these Hemififth > generators makes an octave.
>
> It might also be worth noting that 1 - (2-sqrt(2))/2 = sqrt(2)/2.
>
> > A magic number because the solution of a major fractal > algorithm : x^2 = 4x - 2, or "Persi" (because following the Parsley > leafs progression)
> > It has a P sequence and its own infinite word following the rule :
> > a ->aaab, b->ab, allowing for endless self-similar patterns.
> > MOS cycles are 3, 7, 10, 17, 24, 41, 58, 99, 140, 239...
> > (more or less the same ones as Aksaka-meta temp)
>
> And exactly the ones of hemififths.