Yahoo Tuning Groups Ultimate Backup tuning Daskin Approaching

This is not the name of my new novel, but rather an exciting
announcement. Some of you may have already noticed that the
website of Daskin keyboards, which had been the same teaser
page for something like 10 years, recently changed:

http://daskin.com

This is Paul Vandervoort's company. He plans to ship by the
middle of next year, though he has not given a firm date.
He is taking deposits for the first 5 units now. The pricing
is between a full-size AXiS, and an Opal, Starr Microzone, or
Horvath/Terpstra keyboard.

To my mind, it's pretty much the ideal keyboard, from the
shape of the keys to the textured keytops to the optical
sensors. Paul is arguably the most accomplished Janko
keyboardist living today. He is a professional gigging
musician in the Reno area and has bootstrapped the entire
project.

-Carl
(Disclosure: I did a small amount of paid work on the USB
implementation in 2002. Later, I was given a mechanical
'parasite' keyboard, that mounts over a conventional piano,
which is now with Norman Henry in Colorado.)

🔗hstraub64 <straub@...>

Good news - this is the third time within a relative short period that I can add an entry to the list of generalized keyboards on the xenwiki.

If this one has also the possibility of assigning a separate MIDI channel to each key, then it would be a REALLY ideal keyboard...

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> This is not the name of my new novel, but rather an exciting
> announcement. Some of you may have already noticed that the
> website of Daskin keyboards, which had been the same teaser
> page for something like 10 years, recently changed:
>
> http://daskin.com
>
> This is Paul Vandervoort's company. He plans to ship by the
> middle of next year, though he has not given a firm date.
> He is taking deposits for the first 5 units now. The pricing
> is between a full-size AXiS, and an Opal, Starr Microzone, or
> Horvath/Terpstra keyboard.
>
> To my mind, it's pretty much the ideal keyboard, from the
> shape of the keys to the textured keytops to the optical
> sensors. Paul is arguably the most accomplished Janko
> keyboardist living today. He is a professional gigging
> musician in the Reno area and has bootstrapped the entire
> project.
>
> -Carl
> (Disclosure: I did a small amount of paid work on the USB
> implementation in 2002. Later, I was given a mechanical
> 'parasite' keyboard, that mounts over a conventional piano,
> which is now with Norman Henry in Colorado.)
>

🔗Kalle Aho <kalleaho@...>

🔗Carl Lumma <carl@...>

Hans wrote:

> Good news - this is the third time within a relative short
> period that I can add an entry to the list of generalized
> keyboards on the xenwiki.

And thanks for doing that. Things are really taking off
this year! Even though it's a halberstadt, I'm also very
very excited about the new Infinite Response controller:

http://www.infiniteresponse.com

I haven't played it yet, but it's got all the right
ingredients to be the first MIDI keyboard that doesn't
suck eggs.

> If this one has also the possibility of assigning a
> separate MIDI channel to each key, then it would be
> a REALLY ideal keyboard...

200+ channels? I dunno about that, but every key is
definitely independently addressable.

-Carl

🔗gdsecor <gdsecor@...>

It's not a true generalized keyboard, because the key colors are meaningful only for 12-equal. It's a shame he didn't slant the rows as Bosanquet did, because then the key colors would have been meaningful for any tuning mapped to a single chain of 5ths (and limited only by the number of keys per octave, which appears to be 30 -- aargh! -- just short of 31).

--George

--- In tuning@yahoogroups.com, "hstraub64" <straub@...> wrote:
>
> Good news - this is the third time within a relative short period that I can add an entry to the list of generalized keyboards on the xenwiki.
>
> If this one has also the possibility of assigning a separate MIDI channel to each key, then it would be a REALLY ideal keyboard...
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > This is not the name of my new novel, but rather an exciting
> > announcement. Some of you may have already noticed that the
> > website of Daskin keyboards, which had been the same teaser
> > page for something like 10 years, recently changed:
> >
> > http://daskin.com
> >
> > This is Paul Vandervoort's company. He plans to ship by the
> > middle of next year, though he has not given a firm date.
> > He is taking deposits for the first 5 units now. The pricing
> > is between a full-size AXiS, and an Opal, Starr Microzone, or
> > Horvath/Terpstra keyboard.
> >
> > To my mind, it's pretty much the ideal keyboard, from the
> > shape of the keys to the textured keytops to the optical
> > sensors. Paul is arguably the most accomplished Janko
> > keyboardist living today. He is a professional gigging
> > musician in the Reno area and has bootstrapped the entire
> > project.
> >
> > -Carl
> > (Disclosure: I did a small amount of paid work on the USB
> > implementation in 2002. Later, I was given a mechanical
> > 'parasite' keyboard, that mounts over a conventional piano,
> > which is now with Norman Henry in Colorado.)
> >
>

🔗Carl Lumma <carl@...>

🔗Brofessor <kraiggrady@...>

I also noticed how a scale would run off the keyboard

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:
>
> It's not a true generalized keyboard, because the key colors are meaningful only for 12-equal. It's a shame he didn't slant the rows as Bosanquet did, because then the key colors would have been meaningful for any tuning mapped to a single chain of 5ths (and limited only by the number of keys per octave, which appears to be 30 -- aargh! -- just short of 31).
>
> --George
>
> --- In tuning@yahoogroups.com, "hstraub64" <straub@> wrote:
> >
> > Good news - this is the third time within a relative short period that I can add an entry to the list of generalized keyboards on the xenwiki.
> >
> > If this one has also the possibility of assigning a separate MIDI channel to each key, then it would be a REALLY ideal keyboard...
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > >
> > > This is not the name of my new novel, but rather an exciting
> > > announcement. Some of you may have already noticed that the
> > > website of Daskin keyboards, which had been the same teaser
> > > page for something like 10 years, recently changed:
> > >
> > > http://daskin.com
> > >
> > > This is Paul Vandervoort's company. He plans to ship by the
> > > middle of next year, though he has not given a firm date.
> > > He is taking deposits for the first 5 units now. The pricing
> > > is between a full-size AXiS, and an Opal, Starr Microzone, or
> > > Horvath/Terpstra keyboard.
> > >
> > > To my mind, it's pretty much the ideal keyboard, from the
> > > shape of the keys to the textured keytops to the optical
> > > sensors. Paul is arguably the most accomplished Janko
> > > keyboardist living today. He is a professional gigging
> > > musician in the Reno area and has bootstrapped the entire
> > > project.
> > >
> > > -Carl
> > > (Disclosure: I did a small amount of paid work on the USB
> > > implementation in 2002. Later, I was given a mechanical
> > > 'parasite' keyboard, that mounts over a conventional piano,
> > > which is now with Norman Henry in Colorado.)
> > >
> >
>

🔗gdsecor <gdsecor@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> George wrote:
>
> > It's not a true generalized keyboard, because the key colors
> > are meaningful only for 12-equal.
>
> ?

I meant meaningful only for a 12-tone/octave tuning (not just 12-equal).

Assuming that you map the tones with transpositional invariance, then the color patterns that repeat in the same row for each octave will not hold true for any non-12 tuning such as 17, 19, 22, or extended meantone.

As Kraig also observed, if you use a conventional right-hand fingering for a C major scale (thumbs on C & F, which also *does* maintain a meaningful color pattern, up to a point), then your're going to move diagonally off the edge of the keyboard, at which point you'll have to jump to another row (where the colors are no longer meaningful for a non-12 tuning). This is much less fluid than in a slanted-row Bosanquet arrangement, where you have fingering patterns closer to what you already learned on a Halberstadt keyboard, without having to jump rows.

> > (and limited only by the number of keys per octave, which
> > appears to be 30 -- aargh! -- just short of 31).
>
> The Daskin 6 has 244 keys, the Daskin 5 has 203. That'll
> be > 7 and 6 * 31, respectively. What constraints are you
> placing on the mapping?
>
> -Carl

In this case, I was counting 6 keys/octave * number of rows in the 5-row keyboard (which was the one I saw on the website). Anyway, never mind.

--George

🔗Carl Lumma <carl@...>

Hi George,

> Assuming that you map the tones with transpositional
> invariance, then the color patterns that repeat in the same
> row for each octave will not hold true for any non-12 tuning
> such as 17, 19, 22, or extended meantone.

I guess that depends how you interpret the colors.

Kalle hasn't reported back yet, but he copied me on an
e-mail he sent to Paul. The keytops are not removable,
but there is the possibility of getting a custom color
scheme from the factory.

Colored tape is also a possibility, though admittedly
not ideal.

> As Kraig also observed, if you use a conventional right-
> hand fingering for a C major scale (thumbs on C & F, which
> also *does* maintain a meaningful color pattern, up to
> a point), then your're going to move diagonally off the
> edge of the keyboard,

Erv mentioned this to me as a disadvantage of the Fokker
organ vs. Bosanquet's, in 1998. I brought it up with Paul
in 2001 and he demonstrated a fingering using 3 rows that
doesn't have this problem and which he claims is superior
anyway. It's shown on this page:

http://daskin.com/page5/page5.html

Also, this article explains his choices as far as spacing
between keys, etc.

-Carl

🔗Carl Lumma <carl@...>

🔗Brofessor <kraiggrady@...>

I had looked but did not find what you were referring to..
the fingering of one scale does not give one much when your basic keyboard patterns as Secor (someone who understands then them better than most) pointed out.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > > Paul experience with keyboards doesn't exist. he insisted for
> > > instance the 22 mapped to a standard keyboard worked best if
> >
> > Um, this is Paul Vandervoort, not Paul Erlich.
>
> As you would have seen had you followed the link immediately
> below the text you replied to:
>
> http://daskin.com/page5/page5.html
>
> Furthermore, it's not even true. Paul Erlich has performed
> and recorded with several keyboard players using his mappings
> (including Mike Battaglia less than a month ago), recorded at
> least one track playing them himself, and I saw him play a
> regular piano quite decently also. I'm sure he would prefer a
> generalized keyboard... his mappings are a practical measure.
>
> -Carl
>

🔗Carl Lumma <carl@...>

🔗Brofessor <kraiggrady@...>

I have no problem with this keyboard with a 12 tone tuning.
It is my understanding that the 31 rone organ in Harlem will have the bosanquet layout.
i still need to add this to the archive but
http://anaphoria.com/gralspectrum.pdf
shows how that on a generalized layout following Bonsanquet's lead the generator will alway be the next step up from the line drawn from octave to octave. Here it is applied t o any generator size which is beyond what Bosanquet did.
Perhaps it is trivial that fokkers would be reversed and one could set up a complete mirroring system

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@> wrote:
> >
> > I had looked but did not find what you were referring to..
> > the fingering of one scale does not give one much when your
> > basic keyboard patterns as Secor (someone who understands
> > then them better than most) pointed out.
>
> The fingering is shown here
>
> http://daskin.com/page5/files/keyboard-linkage-etc-adjust-400025.jpg
>
> like any other uniform layout, it works in any key.
>
> I am not saying I would not prefer a Bosanquet arrangement.
> I don't have enough experience with both to tell. But it
> doesn't seem to stop Paul V. from playing up a storm, nor
> did it seem to stop the Dutch 31-toners who used the Fokker
> organ.
>
> -Carl
>

🔗hstraub64 <straub@...>

🔗Graham Breed <gbreed@...>

"Carl Lumma" <carl@...> wrote:

> Furthermore, it's not even true. Paul Erlich has
> performed and recorded with several keyboard players
> using his mappings (including Mike Battaglia less than a
> month ago), recorded at least one track playing them
> himself, and I saw him play a regular piano quite
> decently also. I'm sure he would prefer a generalized
> keyboard... his mappings are a practical measure.

I'll speak up for keyboard mappings that work with the
Halberstadt logic, anyway. I've used Paul Erlich's 24 from
22 for Diaschismic, the 29 note Schismatic (that happens to
be complete), a 24 from 21 Miracle, an 18 from 19 Magic,
and now a 12 from 13 Orwell.

The point of Paul's 24 note Dischismic mapping is that the
black notes give you a decatonic scale. I don't remember
what he said about it for 22-equal in non-decatonic
contexts.

I can speak with authority, because I've been using these
mappings intermittently for over 10 years. I also
comfortably rank among the world's all time top 10 Miracle
and Magic keyboard players to use Halberstadt mappings. By
my own ranking, that is, but why should I give a Mercator's
comma for anybody else's opinion?

Of course, I can appreciate a generalized layout as well.
I used to be the world's number one Miracle on Ztar
player. I'm now reduced to playing the game of "hunt the
pieces I need to get my Ztar working" whenever I stay with
my parents. This consists of them hiding the different
components (the Ztar itself, the power supply, the MIDI
cables, the box you need to connect the power supply to a
MIDI cable, the various synthesizers I could plug it into)
at random locations around the house and not telling me
where.

Graham

🔗Kalle Aho <kalleaho@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi George,
>
> > Assuming that you map the tones with transpositional
> > invariance, then the color patterns that repeat in the same
> > row for each octave will not hold true for any non-12 tuning
> > such as 17, 19, 22, or extended meantone.
>
> I guess that depends how you interpret the colors.
>
> Kalle hasn't reported back yet, but he copied me on an
> e-mail he sent to Paul.

Sorry about that!

> The keytops are not removable, but there is the possibility of
> getting a custom color scheme from the factory.

Yes. I think the textured keytops are a fantastic idea but if one
wants to use the keyboard with many different mappings they
will just confuse the player. :(

Carl, given the chance, what sort of color scheme would you order?

Kalle

🔗Carl Lumma <carl@...>

Kalle wrote:

> Yes. I think the textured keytops are a fantastic idea but if one
> wants to use the keyboard with many different mappings they
> will just confuse the player. :(

It may be possible to think in terms of relationships to
something already learned.

It would certainly also be confusing if you rearranged them.

You know, changing keyboard tunings is also confusing!

> Carl, given the chance, what sort of color scheme would you order?

I probably wouldn't. I imagine I'll try a number of things
rapidly and then settle on one for a long time. It would
be nice to rearrange the key tops at that point, to have a
big bucket of colored tops to choose from, etc. But this
is reality. I'll probably just keep the default pattern
through it all.

I don't like the Wilson/Starr flat honeycomb nearly as much.
If I did, and if the price were the same... that has adhesive
keytops you can rearrange. At least, that's what Starr told
me in 1998. That was before he finished his first unit.

-Carl

🔗Carl Lumma <carl@...>

🔗Brofessor <kraiggrady@...>

the Daskin layout you could put on a honeycomb pattern. one just tilts the keyboard to play. a common thing to do with some of the mappings. One wishes for a set of different keys textured and or colored one could pop on and off like game pieces

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Kalle wrote:
>
> > Yes. I think the textured keytops are a fantastic idea but if one
> > wants to use the keyboard with many different mappings they
> > will just confuse the player. :(
>
> It may be possible to think in terms of relationships to
> something already learned.
>
> It would certainly also be confusing if you rearranged them.
>
> You know, changing keyboard tunings is also confusing!
>
> > Carl, given the chance, what sort of color scheme would you order?
>
> I probably wouldn't. I imagine I'll try a number of things
> rapidly and then settle on one for a long time. It would
> be nice to rearrange the key tops at that point, to have a
> big bucket of colored tops to choose from, etc. But this
> is reality. I'll probably just keep the default pattern
> through it all.
>
> I don't like the Wilson/Starr flat honeycomb nearly as much.
> If I did, and if the price were the same... that has adhesive
> keytops you can rearrange. At least, that's what Starr told
> me in 1998. That was before he finished his first unit.
>
> -Carl
>

🔗Carl Lumma <carl@...>

🔗Kalle Aho <kalleaho@...>

🔗gdsecor <gdsecor@...>

Whatever color scheme you decide on would still be valid only for a fixed number of tones/octave. I would advise keeping the keyboard as is and using a few stick-ons to mark the C's for other tunings. You would be playing mostly by vectors and patterns anyway (i.e., more by touch than by sight), so the color disorientation would be a problem mostly in instances where you have to move your entire hand to make a large jump in pitch.

> Colored tape is also a possibility, though admittedly
> not ideal.
>
> > As Kraig also observed, if you use a conventional right-
> > hand fingering for a C major scale (thumbs on C & F, which
> > also *does* maintain a meaningful color pattern, up to
> > a point), then your're going to move diagonally off the
> > edge of the keyboard,
>
> Erv mentioned this to me as a disadvantage of the Fokker
> organ vs. Bosanquet's, in 1998.

Another disadvantage of the Fokker keyboard is that it isn't generalized, because the key colors are not consistent with tunings having other than 31 tones/octave.

BTW, per a recent message, I saw that Kraig is under the mistaken impression that I came up with the Scalatron generalized keyboard completely from scratch. I already knew about the Fokker keyboard, and I thought that it could be improved by: 1) reversing the order of the rows by mentally flipping it over along the x-axis, and 2) slanting the rows so that octaves of the same pitch were exactly lateral. Only afterward did I realize that the result would accommodate any tuning that could be mapped to a single chain of 5ths.

> I brought it up with Paul
> in 2001 and he demonstrated a fingering using 3 rows that
> doesn't have this problem and which he claims is superior
> anyway. It's shown on this page:
>
> http://daskin.com/page5/page5.html

Yes, that fingering would be very comfortable for the hand, but it would not work for any tuning having other than 12 tones/octave, which further points up the fact that this isn't a generalized keyboard.

--George

🔗Carl Lumma <carl@...>

🔗Brofessor <kraiggrady@...>

Thanks George for filling me in on that history.
It seem 1998 is late considering it was in Xenharmonikon along side some of your other work?

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > Hi George,
> >
> > > Assuming that you map the tones with transpositional
> > > invariance, then the color patterns that repeat in the same
> > > row for each octave will not hold true for any non-12 tuning
> > > such as 17, 19, 22, or extended meantone.
> >
> > I guess that depends how you interpret the colors.
> >
> > Kalle hasn't reported back yet, but he copied me on an
> > e-mail he sent to Paul. The keytops are not removable,
> > but there is the possibility of getting a custom color
> > scheme from the factory.
>
> Whatever color scheme you decide on would still be valid only for a fixed number of tones/octave. I would advise keeping the keyboard as is and using a few stick-ons to mark the C's for other tunings. You would be playing mostly by vectors and patterns anyway (i.e., more by touch than by sight), so the color disorientation would be a problem mostly in instances where you have to move your entire hand to make a large jump in pitch.
>
> > Colored tape is also a possibility, though admittedly
> > not ideal.
> >
> > > As Kraig also observed, if you use a conventional right-
> > > hand fingering for a C major scale (thumbs on C & F, which
> > > also *does* maintain a meaningful color pattern, up to
> > > a point), then your're going to move diagonally off the
> > > edge of the keyboard,
> >
> > Erv mentioned this to me as a disadvantage of the Fokker
> > organ vs. Bosanquet's, in 1998.
>
> Another disadvantage of the Fokker keyboard is that it isn't generalized, because the key colors are not consistent with tunings having other than 31 tones/octave.
>
> BTW, per a recent message, I saw that Kraig is under the mistaken impression that I came up with the Scalatron generalized keyboard completely from scratch. I already knew about the Fokker keyboard, and I thought that it could be improved by: 1) reversing the order of the rows by mentally flipping it over along the x-axis, and 2) slanting the rows so that octaves of the same pitch were exactly lateral. Only afterward did I realize that the result would accommodate any tuning that could be mapped to a single chain of 5ths.
>
> > I brought it up with Paul
> > in 2001 and he demonstrated a fingering using 3 rows that
> > doesn't have this problem and which he claims is superior
> > anyway. It's shown on this page:
> >
> > http://daskin.com/page5/page5.html
>
> Yes, that fingering would be very comfortable for the hand, but it would not work for any tuning having other than 12 tones/octave, which further points up the fact that this isn't a generalized keyboard.
>
> --George
>

🔗gdsecor <gdsecor@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> > > http://daskin.com/page5/page5.html
> >
> > Yes, that fingering would be very comfortable for the hand,
> > but it would not work for any tuning having other than
> > 12 tones/octave, which further points up the fact that this
> > isn't a generalized keyboard.
>
> I have no idea what you mean - could you explain?
>
> -Carl

The fingering pattern assumes that the octave vector is 6 keys to the right, same row as starting tone. Since 1 key to the right is a major 2nd, then this amounts to six major 2nds up. Going up six major 2nds in 17, 19, 22, 31, meantone, pythagorean, or anything else that's not 12 does not take you up an octave; you would have to go right 6 keys and down 2 rows to get to the octave. For any EDO, the octave will also be available in another row (assuming that there are a sufficient number of rows), but its location will vary according to the number of tones/octave: for 17 go up 3 rows and move to the left (the key to the left of the crack); for 19 go up 3 rows and move to the right; for 22 go up 4 rows and 1 key to the left; for 31 go up 5 rows and move to the right. This sort of adjustment will be necessary if you're going much more than an octave away from the region around middle C. In other words, the keys 2 octaves above and below middle C will be in different locations with different numbers of tones/octave.

I was wondering whether my failure to communicate my point is because we have different definitions of the term "generalized". As Bosanquet conceived the first generalilzed keyboard, it is not enough for a keyboard to have transpositional invariance (and I think I'm correct in assuming that you will want to map the pitches of whatever tuning with transpositional invariance). It must also have fingering patterns in common across multiple tunings over the entire range of the keyboard, insofar as those tunings will allow.

There are instances where the fingerings for certain intervals cannot be the same for different tunings, for example, the best approximation of 4:5 in 19-ET vs. 22-ET occurs at different locations in a chain of fifths, so that this interval *must* be played differently on Bosanquet's keyboard. However, octaves, fourths, fifths, and major 2nds, and combinations thereof, are all played alike in those two tunings, so the fingering patterns for those intervals are generalized across those tunings. For 12, 19, and 31-ET (and any other meantone-class tuning) the generalized patterns also include 3rds, 6ths, and many augmented and diminished intervals.

--George

🔗Mike Battaglia <battaglia01@...>

On Thu, Nov 11, 2010 at 8:55 PM, Carl Lumma <carl@...> wrote:
>
> > > Paul experience with keyboards doesn't exist. he insisted for
> > > instance the 22 mapped to a standard keyboard worked best if
> >
> > Um, this is Paul Vandervoort, not Paul Erlich.
>
> As you would have seen had you followed the link immediately
> below the text you replied to:
>
> http://daskin.com/page5/page5.html
>
> Furthermore, it's not even true. Paul Erlich has performed
> and recorded with several keyboard players using his mappings
> (including Mike Battaglia less than a month ago), recorded at
> least one track playing them himself, and I saw him play a
> regular piano quite decently also. I'm sure he would prefer a
> generalized keyboard... his mappings are a practical measure.

Yeah, and it was awesome. I couldn't handle his 22 mapping though - he
had 2 octaves with the E's missing, so it ended up tiling every 2
octaves. I ended up playing in a 12-note subset of porcupine instead
because it was too hard to figure out what was going on with the 22
mapping. He was really good with the 22 keyboard layout thought, he
was improvising Gene Smith-esque style 11-limit harmonies that were
modulating in all kinds of weird ways, and it was #*$(&@#* sweet.

I could probably get used to his 22 layout, since it turns out that
all of the black keys make the SPM major scale (or maybe it was the
symmetrical). So if you treat that like the "diatonic" scale for the
tuning, then you end up getting this kind of inverted Halberstadt
arrangement.

He was also really good at regular piano, so it wasn't that he just
has no experience with keyboards or anything like that.

-Mike

🔗gdsecor <gdsecor@...>

1998 is merely the year that Erv told Carl that the Fokker keyboard had certain disadvantages. My improvements to the Fokker keyboard that resulted in my rediscovery of Bosanquet's generalized keyboard occurred in 1974.

--George

--- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@...> wrote:
>
> Thanks George for filling me in on that history.
> It seem 1998 is late considering it was in Xenharmonikon along side some of your other work?
>
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> > >
> > > Hi George,
> > >
> > > > Assuming that you map the tones with transpositional
> > > > invariance, then the color patterns that repeat in the same
> > > > row for each octave will not hold true for any non-12 tuning
> > > > such as 17, 19, 22, or extended meantone.
> > >
> > > I guess that depends how you interpret the colors.
> > >
> > > Kalle hasn't reported back yet, but he copied me on an
> > > e-mail he sent to Paul. The keytops are not removable,
> > > but there is the possibility of getting a custom color
> > > scheme from the factory.
> >
> > Whatever color scheme you decide on would still be valid only for a fixed number of tones/octave. I would advise keeping the keyboard as is and using a few stick-ons to mark the C's for other tunings. You would be playing mostly by vectors and patterns anyway (i.e., more by touch than by sight), so the color disorientation would be a problem mostly in instances where you have to move your entire hand to make a large jump in pitch.
> >
> > > Colored tape is also a possibility, though admittedly
> > > not ideal.
> > >
> > > > As Kraig also observed, if you use a conventional right-
> > > > hand fingering for a C major scale (thumbs on C & F, which
> > > > also *does* maintain a meaningful color pattern, up to
> > > > a point), then your're going to move diagonally off the
> > > > edge of the keyboard,
> > >
> > > Erv mentioned this to me as a disadvantage of the Fokker
> > > organ vs. Bosanquet's, in 1998.
> >
> > Another disadvantage of the Fokker keyboard is that it isn't generalized, because the key colors are not consistent with tunings having other than 31 tones/octave.
> >
> > BTW, per a recent message, I saw that Kraig is under the mistaken impression that I came up with the Scalatron generalized keyboard completely from scratch. I already knew about the Fokker keyboard, and I thought that it could be improved by: 1) reversing the order of the rows by mentally flipping it over along the x-axis, and 2) slanting the rows so that octaves of the same pitch were exactly lateral. Only afterward did I realize that the result would accommodate any tuning that could be mapped to a single chain of 5ths.
> >
> > > I brought it up with Paul
> > > in 2001 and he demonstrated a fingering using 3 rows that
> > > doesn't have this problem and which he claims is superior
> > > anyway. It's shown on this page:
> > >
> > > http://daskin.com/page5/page5.html
> >
> > Yes, that fingering would be very comfortable for the hand, but it would not work for any tuning having other than 12 tones/octave, which further points up the fact that this isn't a generalized keyboard.
> >
> > --George
> >
>

🔗Mike Battaglia <battaglia01@...>

On Sat, Nov 13, 2010 at 12:17 AM, gdsecor <gdsecor@...> wrote:
>
> The fingering pattern assumes that the octave vector is 6 keys to the right, same row as starting tone. Since 1 key to the right is a major 2nd, then this amounts to six major 2nds up. Going up six major 2nds in 17, 19, 22, 31, meantone, pythagorean, or anything else that's not 12 does not take you up an octave; you would have to go right 6 keys and down 2 rows to get to the octave. For any EDO, the octave will also be available in another row (assuming that there are a sufficient number of rows), but its location will vary according to the number of tones/octave: for 17 go up 3 rows and move to the left (the key to the left of the crack); for 19 go up 3 rows and move to the right; for 22 go up 4 rows and 1 key to the left; for 31 go up 5 rows and move to the right. This sort of adjustment will be necessary if you're going much more than an octave away from the region around middle C. In other words, the keys 2 octaves above and below middle C will be in different locations with different numbers of tones/octave.

I've wondered about this. The whole concept of the Janko keyboard is
like it's two whole tone scales a half step apart. How would this
apply to something like 31-et? Is there some kind of generalized Janko
layout?

It also looks like that every key is linked to the key two rows below
it, so that when you depress one, the other simultaneously depresses
as well.

-Mike

🔗Carl Lumma <carl@...>

Hi George,

> I was wondering whether my failure to communicate my point is
> because we have different definitions of the term "generalized".
> As Bosanquet conceived the first generalilzed keyboard, it is
> not enough for a keyboard to have transpositional invariance
> (and I think I'm correct in assuming that you will want to map
> the pitches of whatever tuning with transpositional invariance).
> It must also have fingering patterns in common across multiple
> tunings over the entire range of the keyboard, insofar as those
> tunings will allow.

Ah, yes indeed. I've been using the term rather loosely. I'll
stop doing that.
But of course Bosanquet's keyboard only has this property for
the first-order systems. And a better way to characterize the
Daskin is that it works for zero-order systems.

> 22-ET occurs at different locations in a chain of fifths,

It being doubly positive. Have you seen Bosanquet's doubly
positive keyboard (designed to accommodate the "Hindoo" system)?

In case anybody wants it, I've made a keyboard blank for
the Daskin 6 and placed it here:

http://lumma.org/temp/Daskin6Shaded.png

or if you order it with all-white keys

http://lumma.org/temp/Daskin6.png

The dimensions are about right if 1 pixel = 0.01".

-Carl

🔗Carl Lumma <carl@...>

🔗gdsecor <gdsecor@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi George,
>
> > I was wondering whether my failure to communicate my point is
> > because we have different definitions of the term "generalized".
> > As Bosanquet conceived the first generalilzed keyboard, it is
> > not enough for a keyboard to have transpositional invariance
> > (and I think I'm correct in assuming that you will want to map
> > the pitches of whatever tuning with transpositional invariance).
> > It must also have fingering patterns in common across multiple
> > tunings over the entire range of the keyboard, insofar as those
> > tunings will allow.
>
> Ah, yes indeed. I've been using the term rather loosely. I'll
> stop doing that.
> But of course Bosanquet's keyboard only has this property for
> the first-order systems.

What do you mean by "first-order systems"?

> And a better way to characterize the
> Daskin is that it works for zero-order systems.
>
> > 22-ET occurs at different locations in a chain of fifths,
>
> It being doubly positive. Have you seen Bosanquet's doubly
> positive keyboard (designed to accommodate the "Hindoo" system)?

The "Hindoo" scale is much closer to Pythagorean[22] than 22-equal, so the (singly) positive stop on his generalized-keyboard organ would have sufficed.

I've read Bosanquet's 1876 "Elementary Treatise on Musical Intervals and Temperament", in which he doesn't even mention the 22 division, although I understand he wrote about 22 later on. The generalized keyboard described in his 1876 treatise will work just fine with any tuning mapped to 22 tones in a chain of 5ths (I know so, because I've used 22-equal many times on my Scalatron), so I don't really understand your question, because there's no need for a different keyboard.

--George

🔗Kraig Grady <kraiggrady@...>

Each keyboard design can be tied to the scale tree, being able to play any scale found underneath as there are options of two or even three. This is something that hit Erv after sometime around seeing Hansons Keyboard ( i speculate) . His designation of say for example the 4/7 keyboard lets one know that it fits in the scale tree between 1/2 and 3/5 and covers the 7/12 (and also the 5/9 scales) in fact everything between 1/2 and 3/5
I remember Erv telling me that Bosanquet had originally submitted a much larger document to the Royal academy but they were not into his ideas at all and censored much by making him cut the size in half.

In xenharmonikon 3 one can see that the 5/4 will not be in the same place in 31 and the 22 tone Indian scale. the distinction being whether the fifth is larger or smaller than 700 as between 41 and 31.
http://anaphoria.com/xen3b.PDF

/^_,',',',_ //^/Kraig Grady_^_,',',',_
Mesotonal Music from:
_'''''''_ ^North/Western Hemisphere:
North American Embassy of Anaphoria Island <http://anaphoria.com/>

_'''''''_^South/Eastern Hemisphere:
Austronesian Outpost of Anaphoria <http://anaphoriasouth.blogspot.com/>

',',',',',',',',',',',',',',',',',',',',',',',',',',',',',

a momentary antenna as i turn to water
this evaporates - an island once again

On 14/11/10 4:18 PM, gdsecor wrote:
>
>
> --- In tuning@yahoogroups.com > <mailto:tuning%40yahoogroups.com>, "Carl Lumma" <carl@...> wrote:
> >
> > Hi George,
> >
> > > I was wondering whether my failure to communicate my point is
> > > because we have different definitions of the term > "generalized".
> > > As Bosanquet conceived the first generalilzed keyboard, it is
> > > not enough for a keyboard to have transpositional invariance
> > > (and I think I'm correct in assuming that you will want to map
> > > the pitches of whatever tuning with transpositional > invariance).
> > > It must also have fingering patterns in common across multiple
> > > tunings over the entire range of the keyboard, insofar as > those
> > > tunings will allow.
> >
> > Ah, yes indeed. I've been using the term rather loosely. I'll
> > stop doing that.
> > But of course Bosanquet's keyboard only has this property for
> > the first-order systems.
>
> What do you mean by "first-order systems"?
>
> > And a better way to characterize the
> > Daskin is that it works for zero-order systems.
> >
> > > 22-ET occurs at different locations in a chain of fifths,
> >
> > It being doubly positive. Have you seen Bosanquet's doubly
> > positive keyboard (designed to accommodate the "Hindoo" system)?
>
> The "Hindoo" scale is much closer to Pythagorean[22] than > 22-equal, so the (singly) positive stop on his > generalized-keyboard organ would have sufficed.
>
> I've read Bosanquet's 1876 "Elementary Treatise on Musical > Intervals and Temperament", in which he doesn't even mention > the 22 division, although I understand he wrote about 22 later > on. The generalized keyboard described in his 1876 treatise > will work just fine with any tuning mapped to 22 tones in a > chain of 5ths (I know so, because I've used 22-equal many > times on my Scalatron), so I don't really understand your > question, because there's no need for a different keyboard.
>
> --George
>
>

🔗Brofessor <kraiggrady@...>

To go back on a point here.....

There is more advantage to the design of the keyboard being two dimensional.
This way one can tilt the keyboard slightly to enable the player to have the octave horizontal regardless of the relationship of the generator to the period.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
>
>
> Hi George,
>
> > I was wondering whether my failure to communicate my point is
> > because we have different definitions of the term "generalized".
> > As Bosanquet conceived the first generalilzed keyboard, it is
> > not enough for a keyboard to have transpositional invariance
> > (and I think I'm correct in assuming that you will want to map
> > the pitches of whatever tuning with transpositional invariance).
> > It must also have fingering patterns in common across multiple
> > tunings over the entire range of the keyboard, insofar as those
> > tunings will allow.
>
> Ah, yes indeed. I've been using the term rather loosely. I'll
> stop doing that.
> But of course Bosanquet's keyboard only has this property for
> the first-order systems. And a better way to characterize the
> Daskin is that it works for zero-order systems.
>
> > 22-ET occurs at different locations in a chain of fifths,
>
> It being doubly positive. Have you seen Bosanquet's doubly
> positive keyboard (designed to accommodate the "Hindoo" system)?
>
> In case anybody wants it, I've made a keyboard blank for
> the Daskin 6 and placed it here:
>
> http://lumma.org/temp/Daskin6Shaded.png
>
> or if you order it with all-white keys
>
> http://lumma.org/temp/Daskin6.png
>
> The dimensions are about right if 1 pixel = 0.01".
>
> -Carl
>

🔗Graham Breed <gbreed@...>

Kraig Grady <kraiggrady@...> wrote:
> Each keyboard design can be tied to the scale tree, being
> able to play any scale found underneath as there are
> options of two or even three. This is something that hit
> Erv after sometime around seeing Hansons Keyboard ( i
> speculate) . His designation of say for example the 4/7
> keyboard lets one know that it fits in the scale tree
> between 1/2 and 3/5 and covers the 7/12 (and also the 5/9
> scales) in fact everything between 1/2 and 3/5 I remember
> Erv telling me that Bosanquet had originally submitted a
> much larger document to the Royal academy but they were
> not into his ideas at all and censored much by making him
> cut the size in half.

You can tie keyboard mappings to the scale tree if you
assume the ratios correspond to generator/period. But
it's usually assumed that the period's an octave. So in
this sense there are some keyboard mappings that aren't tied
to the scale tree.

"Censored" is a loaded term. What you seem to mean is
"edited". There are several Bosanquet documents in
Robert's archive:

http://gfax.ch/files/Literature/music/

Do any of them contain this missing information?

Graham

🔗Graham Breed <gbreed@...>

"gdsecor" <gdsecor@...> wrote:
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@...>
> wrote:

> > It being doubly positive. Have you seen Bosanquet's
> > doubly positive keyboard (designed to accommodate the
> > "Hindoo" system)?
>
> The "Hindoo" scale is much closer to Pythagorean[22] than
> 22-equal, so the (singly) positive stop on his
> generalized-keyboard organ would have sufficed.

The "Hindoo" paper is here:

http://gfax.ch/files/Literature/music/Bosanquet--On%20the%20Hindoo%20Division%20of%20the%20Octave.pdf

I don't see any mention of Pythagorean intonation. I
haven't checked the arithmetic, but it looks like his
"order 2 class 1" defines diaschismic temperament. So
closely that I'll propose re-naming "Diaschismic" as
"Bosanquet".

A first order keyboard does not work with these tunings.
22-equal can be treated as first-order, but remember your
second criterion for a generalized keyboard: "It must also
have fingering patterns in common across multiple tunings
over the entire range of the keyboard, insofar as those
tunings will allow." The fingering pattern for 22 will not
be shared correctly with 46. Order 2 systems really demand
a different keyboard, and Bosanquet showed this. The
diagram is on p. 382. The later description specified c `e
g as a major third, so the note names must be Pythagorean.
You can see that the whole tones like c-d, which will
approximate 9:8, appear to be divided into two equal
semitones. This is a property of order 2 systems that is
not reflected on a first order keyboard. Bosanquet was
right on the money.

> I've read Bosanquet's 1876 "Elementary Treatise on
> Musical Intervals and Temperament", in which he doesn't
> even mention the 22 division, although I understand he
> wrote about 22 later on. The generalized keyboard
> described in his 1876 treatise will work just fine with
> any tuning mapped to 22 tones in a chain of 5ths (I know
> so, because I've used 22-equal many times on my
> Scalatron), so I don't really understand your question,
> because there's no need for a different keyboard.

Of course it works with a chain of fifths, because the
keyboard is based on a chain of fifths. It doesn't work
with tunings that have two distinct chains of fifths. It
doesn't work at all for 34, does it?

Graham

🔗Brofessor <kraiggrady@...>

I am not completely sure what you are saying .
I t sounds like you are saying the scale tree doesn't apply if the period is the octave.
Either way it doesn't matter. If you treat the bottom number as the octave it doesn't matter what size it is. The scale tree is relative. the 4/7 keyboard tells us nothing by that the generator is in the 4th place in a 7 tone scale. He does this so he doesn't have to recalulate the logs relationship of generator to period.
http://wilsonarchives.blogspot.com/2010/10/another-paper-on-mutable-octave.html

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> Kraig Grady <kraiggrady@...> wrote:
> > Each keyboard design can be tied to the scale tree, being
> > able to play any scale found underneath as there are
> > options of two or even three. This is something that hit
> > Erv after sometime around seeing Hansons Keyboard ( i
> > speculate) . His designation of say for example the 4/7
> > keyboard lets one know that it fits in the scale tree
> > between 1/2 and 3/5 and covers the 7/12 (and also the 5/9
> > scales) in fact everything between 1/2 and 3/5 I remember
> > Erv telling me that Bosanquet had originally submitted a
> > much larger document to the Royal academy but they were
> > not into his ideas at all and censored much by making him
> > cut the size in half.
>
> You can tie keyboard mappings to the scale tree if you
> assume the ratios correspond to generator/period. But
> it's usually assumed that the period's an octave. So in
> this sense there are some keyboard mappings that aren't tied
> to the scale tree.
>
> "Censored" is a loaded term. What you seem to mean is
> "edited". There are several Bosanquet documents in
> Robert's archive:
>
> http://gfax.ch/files/Literature/music/
>
> Do any of them contain this missing information?
>
>
> Graham
>

🔗Brofessor <kraiggrady@...>

One way one can sometimes map a scale with more than one chain if each chain is a the same constant structure or an same MOS is to multiply the distance between the keys and insert the other chain in between.
This is how Erv was able to map the 64 tone 3 out of 7 =3-5-7-9-11-13 Euler genus
page 11 and 12 of http://anaphoria.com/Euler.PDF

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> "gdsecor" <gdsecor@...> wrote:
> >
> > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@>
> > wrote:
>
> > > It being doubly positive. Have you seen Bosanquet's
> > > doubly positive keyboard (designed to accommodate the
> > > "Hindoo" system)?
> >
> > The "Hindoo" scale is much closer to Pythagorean[22] than
> > 22-equal, so the (singly) positive stop on his
> > generalized-keyboard organ would have sufficed.
>
> The "Hindoo" paper is here:
>
> http://gfax.ch/files/Literature/music/Bosanquet--On%20the%20Hindoo%20Division%20of%20the%20Octave.pdf
>
> I don't see any mention of Pythagorean intonation. I
> haven't checked the arithmetic, but it looks like his
> "order 2 class 1" defines diaschismic temperament. So
> closely that I'll propose re-naming "Diaschismic" as
> "Bosanquet".
>
> A first order keyboard does not work with these tunings.
> 22-equal can be treated as first-order, but remember your
> second criterion for a generalized keyboard: "It must also
> have fingering patterns in common across multiple tunings
> over the entire range of the keyboard, insofar as those
> tunings will allow." The fingering pattern for 22 will not
> be shared correctly with 46. Order 2 systems really demand
> a different keyboard, and Bosanquet showed this. The
> diagram is on p. 382. The later description specified c `e
> g as a major third, so the note names must be Pythagorean.
> You can see that the whole tones like c-d, which will
> approximate 9:8, appear to be divided into two equal
> semitones. This is a property of order 2 systems that is
> not reflected on a first order keyboard. Bosanquet was
> right on the money.
>
> > I've read Bosanquet's 1876 "Elementary Treatise on
> > Musical Intervals and Temperament", in which he doesn't
> > even mention the 22 division, although I understand he
> > wrote about 22 later on. The generalized keyboard
> > described in his 1876 treatise will work just fine with
> > any tuning mapped to 22 tones in a chain of 5ths (I know
> > so, because I've used 22-equal many times on my
> > Scalatron), so I don't really understand your question,
> > because there's no need for a different keyboard.
>
> Of course it works with a chain of fifths, because the
> keyboard is based on a chain of fifths. It doesn't work
> with tunings that have two distinct chains of fifths. It
> doesn't work at all for 34, does it?
>
>
> Graham
>

🔗Graham Breed <gbreed@...>

"Brofessor" <kraiggrady@...> wrote:
> I am not completely sure what you are saying .
> I t sounds like you are saying the scale tree doesn't
> apply if the period is the octave. Either way it doesn't
> matter. If you treat the bottom number as the octave it
> doesn't matter what size it is. The scale tree is
> relative. the 4/7 keyboard tells us nothing by that the
> generator is in the 4th place in a 7 tone scale. He does
> this so he doesn't have to recalulate the logs
> relationship of generator to period.
> http://wilsonarchives.blogspot.com/2010/10/another-paper-on-mutable-octave.html

I said that if you define the scale tree such that the
period is the octave, it only applies to scales with a
period of an octave. Of course, you can use any period you
like. And, indeed, 4/7 tells you nothing if you don't
define what 4 and 7 refer to.

There was a context here, about generalized keyboards. Any
rank 2 lattice can take any rank 2 tuning. That should
be obvious. But if a keyboard is designed around a certain
interval as an octave, I expect that interval to be an
octave. If not, the keyboard fails to generalize.

What do you mean by "keyboard design"?

Graham

🔗Brofessor <kraiggrady@...>

I am sorry but i am missing your point.
what does it not do that you wish it did?
the octave is simply anything you expect it to be.
The consistent element if that is what you are missing that makes it 'generalized' or isomorphic is that the generator will always be the next key above if you draw a line from you starting point and you period.
That i think is as much as any generalized keyboard can do if you allow any size generator.
--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> "Brofessor" <kraiggrady@...> wrote:
> > I am not completely sure what you are saying .
> > I t sounds like you are saying the scale tree doesn't
> > apply if the period is the octave. Either way it doesn't
> > matter. If you treat the bottom number as the octave it
> > doesn't matter what size it is. The scale tree is
> > relative. the 4/7 keyboard tells us nothing by that the
> > generator is in the 4th place in a 7 tone scale. He does
> > this so he doesn't have to recalulate the logs
> > relationship of generator to period.
> > http://wilsonarchives.blogspot.com/2010/10/another-paper-on-mutable-octave.html
>
> I said that if you define the scale tree such that the
> period is the octave, it only applies to scales with a
> period of an octave. Of course, you can use any period you
> like. And, indeed, 4/7 tells you nothing if you don't
> define what 4 and 7 refer to.
>
> There was a context here, about generalized keyboards. Any
> rank 2 lattice can take any rank 2 tuning. That should
> be obvious. But if a keyboard is designed around a certain
> interval as an octave, I expect that interval to be an
> octave. If not, the keyboard fails to generalize.
>
> What do you mean by "keyboard design"?
>
>
> Graham
>

🔗Graham Breed <gbreed@...>

"Brofessor" <kraiggrady@...> wrote:
> I am sorry but i am missing your point.

My point was that the period is not always an octave. And
you must have understood it because you paraphrased it.
But you keep asking me questions to keep the thread going.

> what does it not do that you wish it did?

I wish it were generalized, in the sense George defined.

> the octave is simply anything you expect it to be.
> The consistent element if that is what you are missing
> that makes it 'generalized' or isomorphic is that the
> generator will always be the next key above if you draw
> a line from you starting point and you period. That i
> think is as much as any generalized keyboard can do if
> you allow any size generator.

I'm not missing that. I understand that perfectly well.
And it is possible to do better. You can define your
generators to be a major third and perfect fifth. You can
then map any regular temperament that's free of 5-limit
contorsion to the keyboard, and know how to finger major
and minor triads. With a given number of notes to the
octave (constant structure, periodicity block, or
whatever), you even get consistent fingering of octaves,
and so all 5-limit intervals. Because the tunings
wouldn't depart that far from 5-limit JI, basic melodic
patterns (once you manage to locate them) would be
preserved.

As I understand it, this is how the Axis 49, or some
commercial keyboard, works. The standard mapping and key
colors are based on 12, of course. But a 19 note version
would give either Meantone, Magic, or Hanson, with
black/white coloring based around Meantone, and so standard
notation. All you need to change are the key colors and
the MIDI mapping. A 31 note monstrosity would give
Meantone, Miracle, or Orwell. Other numbers would work but
the black/white coloring would lose its meantone analogy.

Graham

🔗Brofessor <kraiggrady@...>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> I wish it were generalized, in the sense George defined.

that the octave and the say a given ratio was always in the same place?
well that might be possible but what happens if you have a tuning that has nether which is what 72% of the scale tree maps? or scales which are non octave?

Having instruments based on 22 and 31 constant structures never once did i worry about the octave not being in the same place. Nor did any of the players i worked with. In fact i found having different keyboard layouts a great advantage being able to play somethings easier on some than others.
Ideally I would prefer many alternate keyboards for the same tuning ,

All these below are easily mapped onto a 7/12 keyboard

>
> As I understand it, this is how the Axis 49, or some
> commercial keyboard, works. The standard mapping and key
> colors are based on 12, of course. But a 19 note version
> would give either Meantone, Magic, or Hanson, with
> black/white coloring based around Meantone, and so standard
> notation. All you need to change are the key colors and
> the MIDI mapping. A 31 note monstrosity would give
> Meantone, Miracle, or Orwell. Other numbers would work but
> the black/white coloring would lose its meantone analogy.

>
>
> Graham
>

🔗Graham Breed <gbreed@...>

"Brofessor" <kraiggrady@...> wrote:
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...>
> wrote:
>
> > I wish it were generalized, in the sense George defined.
>
> that the octave and the say a given ratio was always in
> the same place? well that might be possible but what
> happens if you have a tuning that has nether which is
> what 72% of the scale tree maps? or scales which are non
> octave?

What George said is ". . . it is not enough for a keyboard
to have transpositional invariance . . . It must
also have fingering patterns in common across multiple
tunings over the entire range of the keyboard, insofar as
those tunings will allow." That's what this thread is
about. I don't have anything new to say about it.

Why are you bringing up the scale tree again? I don't
choose scales randomly from it. But if the scale tree maps
ratios, isn't every scale (or rank 2 scale family, extended
to an infinite number of notes) on there somewhere?

And if the scale tree maps ratios, you should say "ratios"
instead of "what the scale tree maps".

> Having instruments based on 22 and 31 constant structures
> never once did i worry about the octave not being in the
> same place. Nor did any of the players i worked with. In
> fact i found having different keyboard layouts a great
> advantage being able to play somethings easier on some
> than others. Ideally I would prefer many alternate
> keyboards for the same tuning ,

It seems you don't want a generalized keyboard then. Good
for you! Why are you bringing your opinions into a thread
about generalized keyboards?

> All these below are easily mapped onto a 7/12 keyboard

Show me how to map Miracle.

Graham

🔗Carl Lumma <carl@...>

George wrote:

> > But of course Bosanquet's keyboard only has this property
> > for the first-order systems.
>
> What do you mean by "first-order systems"?

Bosanquet's terminology, measuring the size of the Pythagorean
comma in steps of the scale (since the keyboard has 12 columns).
He designed a separate keyboard for second-order systems,
where each column of keys has twice the vertical offset.
The Daskin keyboard is a zero-order keyboard. So they're all
generalized in the sense that fingering patterns don't change
for systems of the appropriate order. The Bosanquet designs
have the advantage of handling both positive and negative
systems of the given order, since the keyboard can be mirrored
about its oblong axis. The Daskin doesn't have that advantage
because negative zero is still zero.

> > It being doubly positive. Have you seen Bosanquet's doubly
> > positive keyboard (designed to accommodate the "Hindoo" system)?
>
> The "Hindoo" scale is much closer to Pythagorean[22] than
> 22-equal,

Bosanquet didn't seem to know this (or thought 22-ET was
close enough).

> I've read Bosanquet's 1876 "Elementary Treatise on Musical
> Intervals and Temperament", in which he doesn't even mention
> the 22 division, although I understand he wrote about
> 22 later on. The generalized keyboard described in his 1876
> treatise will work just fine with any tuning mapped
> to 22 tones in a chain of 5ths (I know so, because I've used
> 22-equal many times on my Scalatron), so I don't really
> understand your question, because there's no need for a
> different keyboard.

I don't think you can map it to keep the same fingering
for the consonances and keep the octaves parallel to the
player.

-Carl

🔗Brofessor <kraiggrady@...>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> "Brofessor" <kraiggrady@...> wrote:

>
> What George said is ". . . it is not enough for a keyboard
> to have transpositional invariance . . . It must
> also have fingering patterns in common across multiple
> tunings over the entire range of the keyboard, insofar as
> those tunings will allow." That's what this thread is
> about. I don't have anything new to say about it.
>
> Why are you bringing up the scale tree again? I don't
> choose scales randomly from it. But if the scale tree maps
> ratios, isn't every scale (or rank 2 scale family, extended
> to an infinite number of notes) on there somewhere?

exactly
>
> And if the scale tree maps ratios, you should say "ratios"
> instead of "what the scale tree maps".

the scale tree when using keyboards is not mapping ratios.

>
> > Having instruments based on 22 and 31 constant structures
> > never once did i worry about the octave not being in the
> > same place. Nor did any of the players i worked with. In
> > fact i found having different keyboard layouts a great
> > advantage being able to play somethings easier on some
> > than others. Ideally I would prefer many alternate
> > keyboards for the same tuning ,
>
> It seems you don't want a generalized keyboard then. Good
> for you! Why are you bringing your opinions into a thread
> about generalized keyboards?

these are generalized according to how everyone has been using the term since 1975 in xenharmonikon.

there is no such thing as a keyboard that will keep the octave and map all scales simple .
so the term generalized does not apply just because one person wants the impossible.
there is a keyboard that will map all scales and put the generator directly above, that is why it is called generalized
>
> > All these below are easily mapped onto a 7/12 keyboard
>
> Show me how to map Miracle.
this is a 1/10 keyboard same as Secors.
it is included in the overall generalized pattern

>
>
> Graham
>

🔗Graham Breed <gbreed@...>

"Brofessor" <kraiggrady@...> wrote:
>
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...>
> wrote:

> > Why are you bringing up the scale tree again? I don't
> > choose scales randomly from it. But if the scale tree
> > maps ratios, isn't every scale (or rank 2 scale family,
> > extended to an infinite number of notes) on there
> > somewhere?
>
> exactly

Isn't it on there everywhere, I should have said. Each
ratio (that is each node on the scale tree) will be a
ratio of generators for each MOS family, provided you have
a free choice of generators.

> > And if the scale tree maps ratios, you should say
> > "ratios" instead of "what the scale tree maps".
>
> the scale tree when using keyboards is not mapping
> ratios.

Then why did you say "exactly" when I said it was above?
And what does it map?

> these are generalized according to how everyone has been
> using the term since 1975 in xenharmonikon.

What are? You mean you agree with George's definition?

> there is no such thing as a keyboard that will keep the
> octave and map all scales simple . so the term
> generalized does not apply just because one person wants
> the impossible. there is a keyboard that will map all
> scales and put the generator directly above, that is why
> it is called generalized

No, the generation applies the way George defined it. And
there's no keyboard that will do it for all scales, as you
said at the start.

> > > All these below are easily mapped onto a 7/12 keyboard
> >
> > Show me how to map Miracle.
> this is a 1/10 keyboard same as Secors.
> it is included in the overall generalized pattern

How can 1/10 be 7/12? Secor's Miracle keyboard layout
looks very different to Bosanquet's 7/12.

Graham

🔗Brofessor <kraiggrady@...>

Yes the 1/10 keyboard is the same as Georges, except for the shape of the key.
You are right it not being 7/12.
But when you asked for Miracle i merely calculated the generator and looked at the scale tree and there it was. George's keyboard is apart of what i am referring to as generalized.
Every keyboard will be generalized within a spectrum of different scales.

I think that George found a keyboard in this area that does as he mentions in his article is
brilliant and i don't mean to take that away from him. and if that is all the material i was going to play that might be the keyboard i use.
I think it runs into trouble one you 31 and 41 become constant structures that are ratio based. But i don't think that is a so much of a problem not to contradict myself.
The octave would be the same even if the low to high would reverse it self. That the pattern differs might actually make it easier not to confuse one from the other.
BTW a 1/7 keyboard will also work but I think George's choice is better. I assume he had tried that before the one he illustrates.

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> "Brofessor" <kraiggrady@...> wrote:
> >
> >
> > --- In tuning@yahoogroups.com, Graham Breed <gbreed@>
> > wrote:
>
> > > Why are you bringing up the scale tree again? I don't
> > > choose scales randomly from it. But if the scale tree
> > > maps ratios, isn't every scale (or rank 2 scale family,
> > > extended to an infinite number of notes) on there
> > > somewhere?
> >
> > exactly
>
> Isn't it on there everywhere, I should have said. Each
> ratio (that is each node on the scale tree) will be a
> ratio of generators for each MOS family, provided you have
> a free choice of generators.
>
> > > And if the scale tree maps ratios, you should say
> > > "ratios" instead of "what the scale tree maps".
> >
> > the scale tree when using keyboards is not mapping
> > ratios.
>
> Then why did you say "exactly" when I said it was above?
> And what does it map?
>
> > these are generalized according to how everyone has been
> > using the term since 1975 in xenharmonikon.
>
> What are? You mean you agree with George's definition?
>
> > there is no such thing as a keyboard that will keep the
> > octave and map all scales simple . so the term
> > generalized does not apply just because one person wants
> > the impossible. there is a keyboard that will map all
> > scales and put the generator directly above, that is why
> > it is called generalized
>
> No, the generation applies the way George defined it. And
> there's no keyboard that will do it for all scales, as you
> said at the start.
>
> > > > All these below are easily mapped onto a 7/12 keyboard
> > >
> > > Show me how to map Miracle.
> > this is a 1/10 keyboard same as Secors.
> > it is included in the overall generalized pattern
>
> How can 1/10 be 7/12? Secor's Miracle keyboard layout
> looks very different to Bosanquet's 7/12.
>
>
> Graham
>

🔗gdsecor <gdsecor@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> George wrote:
>
> > > But of course Bosanquet's keyboard only has this property
> > > for the first-order systems.
> >
> > What do you mean by "first-order systems"?
>
> Bosanquet's terminology, measuring the size of the Pythagorean
> comma in steps of the scale (since the keyboard has 12 columns).
> He designed a separate keyboard for second-order systems,
> where each column of keys has twice the vertical offset.
> The Daskin keyboard is a zero-order keyboard. So they're all
> generalized in the sense that fingering patterns don't change
> for systems of the appropriate order. The Bosanquet designs
> have the advantage of handling both positive and negative
> systems of the given order, since the keyboard can be mirrored
> about its oblong axis. The Daskin doesn't have that advantage
> because negative zero is still zero.
>
> > > It being doubly positive. Have you seen Bosanquet's doubly
> > > positive keyboard (designed to accommodate the "Hindoo" system)?
> >
> > The "Hindoo" scale is much closer to Pythagorean[22] than
> > 22-equal,
>
> Bosanquet didn't seem to know this (or thought 22-ET was
> close enough).
>
> > I've read Bosanquet's 1876 "Elementary Treatise on Musical
> > Intervals and Temperament", in which he doesn't even mention
> > the 22 division, although I understand he wrote about
> > 22 later on. The generalized keyboard described in his 1876
> > treatise will work just fine with any tuning mapped
> > to 22 tones in a chain of 5ths (I know so, because I've used
> > 22-equal many times on my Scalatron), so I don't really
> > understand your question, because there's no need for a
> > different keyboard.
>
> I don't think you can map it to keep the same fingering
> for the consonances and keep the octaves parallel to the
> player.

Only the 3-limit consonances will map alike for 22, 41, and 31 (comparing doubly-positive 22 with typical divisions that are singly positive and singly negative, respectively), and the same holds true even if you take 22 out of the picture. For example, 5/4 of C is E in 31, Fb (or E\) in 41, and D# (or E\) in 22 (noting that even E\ map to different physical keys in 41 and 22). But the Bosanquet keyboard is still generalized *insofar* as the tunings permit the same fingerings, and the octaves in 22 are still exactly lateral.

My point is that Bosanquet's original generalized keyboard works just fine for 22; you merely have to learn different locations for primes 5, 7, and 11. The fact that there are two different fingering vectors for a 4:5 does not contradict the fact that there is still transpositional invariance; these are merely duplicate keys that provide alternative fingerings. These alternates are available in every key (as long as a sufficient number of duplicate keys are provided), and they make fingering much easier, especially for the thumb.

I can't see any point in having another generalized keyboard for 22, because with a different keyboard you would also have to learn new fingering patterns.

Ah, it just occured to me that if you want the 34 or 58 divisions, for example, you'll need a different keyboard, which will also do 22, 46, and 56. I'd call this a pajara keyboard. See:
/tuning-math/files/secor/kbds/
The files beginning with "KbPaj" are the pajara keyboards; the 3 are essentially alike, differing only in the accidentals shown (which are valid for both divisions in each diagram).

--George

🔗Brofessor <kraiggrady@...>

is it possible to move a copy of the file here for those not members of tuning math?
or if someone can forward a copy?
Xenharmonikon 3 illustrates the different harmonic templates of these various systems being discussed. Perhaps the only thing one might mention is that while 5/4 will move to a new place on the keyboard, it is often replace with another "fifth' such as a 81/64 when the tuning allows comma differences.

--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:
>
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > George wrote:
> >
> > > > But of course Bosanquet's keyboard only has this property
> > > > for the first-order systems.
> > >
> > > What do you mean by "first-order systems"?
> >
> > Bosanquet's terminology, measuring the size of the Pythagorean
> > comma in steps of the scale (since the keyboard has 12 columns).
> > He designed a separate keyboard for second-order systems,
> > where each column of keys has twice the vertical offset.
> > The Daskin keyboard is a zero-order keyboard. So they're all
> > generalized in the sense that fingering patterns don't change
> > for systems of the appropriate order. The Bosanquet designs
> > have the advantage of handling both positive and negative
> > systems of the given order, since the keyboard can be mirrored
> > about its oblong axis. The Daskin doesn't have that advantage
> > because negative zero is still zero.
> >
> > > > It being doubly positive. Have you seen Bosanquet's doubly
> > > > positive keyboard (designed to accommodate the "Hindoo" system)?
> > >
> > > The "Hindoo" scale is much closer to Pythagorean[22] than
> > > 22-equal,
> >
> > Bosanquet didn't seem to know this (or thought 22-ET was
> > close enough).
> >
> > > I've read Bosanquet's 1876 "Elementary Treatise on Musical
> > > Intervals and Temperament", in which he doesn't even mention
> > > the 22 division, although I understand he wrote about
> > > 22 later on. The generalized keyboard described in his 1876
> > > treatise will work just fine with any tuning mapped
> > > to 22 tones in a chain of 5ths (I know so, because I've used
> > > 22-equal many times on my Scalatron), so I don't really
> > > understand your question, because there's no need for a
> > > different keyboard.
> >
> > I don't think you can map it to keep the same fingering
> > for the consonances and keep the octaves parallel to the
> > player.
>
> Only the 3-limit consonances will map alike for 22, 41, and 31 (comparing doubly-positive 22 with typical divisions that are singly positive and singly negative, respectively), and the same holds true even if you take 22 out of the picture. For example, 5/4 of C is E in 31, Fb (or E\) in 41, and D# (or E\) in 22 (noting that even E\ map to different physical keys in 41 and 22). But the Bosanquet keyboard is still generalized *insofar* as the tunings permit the same fingerings, and the octaves in 22 are still exactly lateral.
>
> My point is that Bosanquet's original generalized keyboard works just fine for 22; you merely have to learn different locations for primes 5, 7, and 11. The fact that there are two different fingering vectors for a 4:5 does not contradict the fact that there is still transpositional invariance; these are merely duplicate keys that provide alternative fingerings. These alternates are available in every key (as long as a sufficient number of duplicate keys are provided), and they make fingering much easier, especially for the thumb.
>
> I can't see any point in having another generalized keyboard for 22, because with a different keyboard you would also have to learn new fingering patterns.
>
> Ah, it just occured to me that if you want the 34 or 58 divisions, for example, you'll need a different keyboard, which will also do 22, 46, and 56. I'd call this a pajara keyboard. See:
> /tuning-math/files/secor/kbds/
> The files beginning with "KbPaj" are the pajara keyboards; the 3 are essentially alike, differing only in the accidentals shown (which are valid for both divisions in each diagram).
>
> --George
>

🔗gdsecor <gdsecor@...>

Thanks for locating that. This is the same keyboard geometry as the pajara keyboard diagrams I referred to in my previous message:
/tuning/topicId_94391.html#94614

> I don't see any mention of Pythagorean intonation. I
> haven't checked the arithmetic, but it looks like his
> "order 2 class 1" defines diaschismic temperament. So
> closely that I'll propose re-naming "Diaschismic" as
> "Bosanquet".

I'll have to take your word for it, because I haven't yet read the paper. I only had time to look at the keyboard diagram, which instantly showed me that it's essentially a pajara keyboard. The vertically aligned keys in my diagrams are different from Bosanquet's; mine differ by 1 degree of 34 and 56 and are duplicates in 22.

> A first order keyboard does not work with these tunings.
> 22-equal can be treated as first-order, but remember your
> second criterion for a generalized keyboard: "It must also
> have fingering patterns in common across multiple tunings
> over the entire range of the keyboard, insofar as those
> tunings will allow."

The original generalized keyboard will work with second-order tunings that have all the tones in a single chain of 5ths, in other words, about 1/2 of them. Thus 22 and 46 will work, but 34 and 58 won't. This keyboard will also work with 56 (probably the optimal division for pajara), which is third-order.

> The fingering pattern for 22 will not
> be shared correctly with 46. Order 2 systems really demand
> a different keyboard, and Bosanquet showed this. The
> diagram is on p. 382. The later description specified c `e
> g as a major third, so the note names must be Pythagorean.
> You can see that the whole tones like c-d, which will
> approximate 9:8, appear to be divided into two equal
> semitones. This is a property of order 2 systems that is
> not reflected on a first order keyboard. Bosanquet was
> right on the money.

I don't think he got it exactly right. The suitability of the keyboard depends on the number of chains of fifths, not on the order of the division (as Bosanquet defined it).

Furthermore, on the first-order keyboard the positive first-order divisions don't have primes 5, 7, and 11 mapped the same way as the negative first-order divisions (yet this doesn't disqualify the keyboard from being generalized). It's all a matter of "insofar as the tunings allow it." If you wanted 31 and 41 to have all the primes mapped the same way, you'd need a decimal (Miracle) keyboard, which would also cover 72. (The decimal keyboard will also accommodate 19 and 53, but the mapping of most of the primes is quite different.)

> > I've read Bosanquet's 1876 "Elementary Treatise on
> > Musical Intervals and Temperament", in which he doesn't
> > even mention the 22 division, although I understand he
> > wrote about 22 later on. The generalized keyboard
> > described in his 1876 treatise will work just fine with
> > any tuning mapped to 22 tones in a chain of 5ths (I know
> > so, because I've used 22-equal many times on my
> > Scalatron), so I don't really understand your question,
> > because there's no need for a different keyboard.
>
> Of course it works with a chain of fifths, because the
> keyboard is based on a chain of fifths. It doesn't work
> with tunings that have two distinct chains of fifths. It
> doesn't work at all for 34, does it?

No, it doesn't. Chains of fifths, not Bosanquet-order, is what determines the suitability of the keyboard.

No one keyboard will do it all, but Bosanquet's original generalized keyboard does more than any other.

--George

🔗Carl Lumma <carl@...>

Hi George,

> Only the 3-limit consonances will map alike for 22, 41, and 31
> (comparing doubly-positive 22 with typical divisions that are
> singly positive and singly negative, respectively)

Yes.

> But the Bosanquet keyboard is still generalized *insofar* as
> the tunings permit the same fingerings, and the octaves in 22
> are still exactly lateral.

Fingerings for... the chain of 5ths, but not much else.

> My point is that Bosanquet's original generalized keyboard
> works just fine for 22; you merely have to learn different
> locations for primes 5, 7, and 11. The fact that there are
> two different fingering vectors for a 4:5 does not contradict
> the fact that there is still transpositional invariance;

Sure.

> I can't see any point in having another generalized keyboard
> for 22, because with a different keyboard you would also have
> to learn new fingering patterns.

It seems to me new patterns must be learnt either way.
But I might agree that a new arrangement of digitals is
not called for. I have to do more experimentation with it.

> Ah, it just occured to me that if you want the 34 or 58
> divisions, for example, you'll need a different keyboard,
> which will also do 22, 46, and 56. I'd call this a pajara
> keyboard. See:
> /tuning-math/files/secor/kbds/

Wow, a 22-column layout. Interesting. I think I'd prefer
something more compact (more along the lines of the 5-column
layout recently suggested by Kalle) but I'll have to do more
experimenting.

-Carl

🔗Brofessor <kraiggrady@...>

I was not able to figure out what first order meant from Carl's description.
can you clarify
--- In tuning@yahoogroups.com, "gdsecor" <gdsecor@...> wrote:
>
>
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> >
> > "gdsecor" <gdsecor@> wrote:
> > >
> > > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@>
> > > wrote:
> >
> > > > It being doubly positive. Have you seen Bosanquet's
> > > > doubly positive keyboard (designed to accommodate the
> > > > "Hindoo" system)?
> > >
> > > The "Hindoo" scale is much closer to Pythagorean[22] than
> > > 22-equal, so the (singly) positive stop on his
> > > generalized-keyboard organ would have sufficed.
> >
> > The "Hindoo" paper is here:
> >
> > http://gfax.ch/files/Literature/music/Bosanquet--On%20the%20Hindoo%20Division%20of%20the%20Octave.pdf
>
> Thanks for locating that. This is the same keyboard geometry as the pajara keyboard diagrams I referred to in my previous message:
> /tuning/topicId_94391.html#94614
>
> > I don't see any mention of Pythagorean intonation. I
> > haven't checked the arithmetic, but it looks like his
> > "order 2 class 1" defines diaschismic temperament. So
> > closely that I'll propose re-naming "Diaschismic" as
> > "Bosanquet".
>
> I'll have to take your word for it, because I haven't yet read the paper. I only had time to look at the keyboard diagram, which instantly showed me that it's essentially a pajara keyboard. The vertically aligned keys in my diagrams are different from Bosanquet's; mine differ by 1 degree of 34 and 56 and are duplicates in 22.
>
> > A first order keyboard does not work with these tunings.
> > 22-equal can be treated as first-order, but remember your
> > second criterion for a generalized keyboard: "It must also
> > have fingering patterns in common across multiple tunings
> > over the entire range of the keyboard, insofar as those
> > tunings will allow."
>
> The original generalized keyboard will work with second-order tunings that have all the tones in a single chain of 5ths, in other words, about 1/2 of them. Thus 22 and 46 will work, but 34 and 58 won't. This keyboard will also work with 56 (probably the optimal division for pajara), which is third-order.
>
> > The fingering pattern for 22 will not
> > be shared correctly with 46. Order 2 systems really demand
> > a different keyboard, and Bosanquet showed this. The
> > diagram is on p. 382. The later description specified c `e
> > g as a major third, so the note names must be Pythagorean.
> > You can see that the whole tones like c-d, which will
> > approximate 9:8, appear to be divided into two equal
> > semitones. This is a property of order 2 systems that is
> > not reflected on a first order keyboard. Bosanquet was
> > right on the money.
>
> I don't think he got it exactly right. The suitability of the keyboard depends on the number of chains of fifths, not on the order of the division (as Bosanquet defined it).
>
> Furthermore, on the first-order keyboard the positive first-order divisions don't have primes 5, 7, and 11 mapped the same way as the negative first-order divisions (yet this doesn't disqualify the keyboard from being generalized). It's all a matter of "insofar as the tunings allow it." If you wanted 31 and 41 to have all the primes mapped the same way, you'd need a decimal (Miracle) keyboard, which would also cover 72. (The decimal keyboard will also accommodate 19 and 53, but the mapping of most of the primes is quite different.)
>
> > > I've read Bosanquet's 1876 "Elementary Treatise on
> > > Musical Intervals and Temperament", in which he doesn't
> > > even mention the 22 division, although I understand he
> > > wrote about 22 later on. The generalized keyboard
> > > described in his 1876 treatise will work just fine with
> > > any tuning mapped to 22 tones in a chain of 5ths (I know
> > > so, because I've used 22-equal many times on my
> > > Scalatron), so I don't really understand your question,
> > > because there's no need for a different keyboard.
> >
> > Of course it works with a chain of fifths, because the
> > keyboard is based on a chain of fifths. It doesn't work
> > with tunings that have two distinct chains of fifths. It
> > doesn't work at all for 34, does it?
>
> No, it doesn't. Chains of fifths, not Bosanquet-order, is what determines the suitability of the keyboard.
>
> No one keyboard will do it all, but Bosanquet's original generalized keyboard does more than any other.
>
> --George
>

🔗Brofessor <kraiggrady@...>

found it on page 377. a process i mentioned earlier

--- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@...> wrote:
>
> I was not able to figure out what first order meant from Carl's description.
> can you clarify
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@> wrote:
> >
> >
> >
> > --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
> > >
> > > "gdsecor" <gdsecor@> wrote:
> > > >
> > > > --- In tuning@yahoogroups.com, "Carl Lumma" <carl@>
> > > > wrote:
> > >
> > > > > It being doubly positive. Have you seen Bosanquet's
> > > > > doubly positive keyboard (designed to accommodate the
> > > > > "Hindoo" system)?
> > > >
> > > > The "Hindoo" scale is much closer to Pythagorean[22] than
> > > > 22-equal, so the (singly) positive stop on his
> > > > generalized-keyboard organ would have sufficed.
> > >
> > > The "Hindoo" paper is here:
> > >
> > > http://gfax.ch/files/Literature/music/Bosanquet--On%20the%20Hindoo%20Division%20of%20the%20Octave.pdf
> >
> > Thanks for locating that. This is the same keyboard geometry as the pajara keyboard diagrams I referred to in my previous message:
> > /tuning/topicId_94391.html#94614
> >
> > > I don't see any mention of Pythagorean intonation. I
> > > haven't checked the arithmetic, but it looks like his
> > > "order 2 class 1" defines diaschismic temperament. So
> > > closely that I'll propose re-naming "Diaschismic" as
> > > "Bosanquet".
> >
> > I'll have to take your word for it, because I haven't yet read the paper. I only had time to look at the keyboard diagram, which instantly showed me that it's essentially a pajara keyboard. The vertically aligned keys in my diagrams are different from Bosanquet's; mine differ by 1 degree of 34 and 56 and are duplicates in 22.
> >
> > > A first order keyboard does not work with these tunings.
> > > 22-equal can be treated as first-order, but remember your
> > > second criterion for a generalized keyboard: "It must also
> > > have fingering patterns in common across multiple tunings
> > > over the entire range of the keyboard, insofar as those
> > > tunings will allow."
> >
> > The original generalized keyboard will work with second-order tunings that have all the tones in a single chain of 5ths, in other words, about 1/2 of them. Thus 22 and 46 will work, but 34 and 58 won't. This keyboard will also work with 56 (probably the optimal division for pajara), which is third-order.
> >
> > > The fingering pattern for 22 will not
> > > be shared correctly with 46. Order 2 systems really demand
> > > a different keyboard, and Bosanquet showed this. The
> > > diagram is on p. 382. The later description specified c `e
> > > g as a major third, so the note names must be Pythagorean.
> > > You can see that the whole tones like c-d, which will
> > > approximate 9:8, appear to be divided into two equal
> > > semitones. This is a property of order 2 systems that is
> > > not reflected on a first order keyboard. Bosanquet was
> > > right on the money.
> >
> > I don't think he got it exactly right. The suitability of the keyboard depends on the number of chains of fifths, not on the order of the division (as Bosanquet defined it).
> >
> > Furthermore, on the first-order keyboard the positive first-order divisions don't have primes 5, 7, and 11 mapped the same way as the negative first-order divisions (yet this doesn't disqualify the keyboard from being generalized). It's all a matter of "insofar as the tunings allow it." If you wanted 31 and 41 to have all the primes mapped the same way, you'd need a decimal (Miracle) keyboard, which would also cover 72. (The decimal keyboard will also accommodate 19 and 53, but the mapping of most of the primes is quite different.)
> >
> > > > I've read Bosanquet's 1876 "Elementary Treatise on
> > > > Musical Intervals and Temperament", in which he doesn't
> > > > even mention the 22 division, although I understand he
> > > > wrote about 22 later on. The generalized keyboard
> > > > described in his 1876 treatise will work just fine with
> > > > any tuning mapped to 22 tones in a chain of 5ths (I know
> > > > so, because I've used 22-equal many times on my
> > > > Scalatron), so I don't really understand your question,
> > > > because there's no need for a different keyboard.
> > >
> > > Of course it works with a chain of fifths, because the
> > > keyboard is based on a chain of fifths. It doesn't work
> > > with tunings that have two distinct chains of fifths. It
> > > doesn't work at all for 34, does it?
> >
> > No, it doesn't. Chains of fifths, not Bosanquet-order, is what determines the suitability of the keyboard.
> >
> > No one keyboard will do it all, but Bosanquet's original generalized keyboard does more than any other.
> >
> > --George
> >
>

🔗Brofessor <kraiggrady@...>

This keyboard is a 7/12 keyboard on the scale tree which is expanded from the 4/7 keyboard directly above it.

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> Hi George,
>
> > Only the 3-limit consonances will map alike for 22, 41, and 31
> > (comparing doubly-positive 22 with typical divisions that are
> > singly positive and singly negative, respectively)
>
> Yes.
>
> > But the Bosanquet keyboard is still generalized *insofar* as
> > the tunings permit the same fingerings, and the octaves in 22
> > are still exactly lateral.
>
> Fingerings for... the chain of 5ths, but not much else.
>
> > My point is that Bosanquet's original generalized keyboard
> > works just fine for 22; you merely have to learn different
> > locations for primes 5, 7, and 11. The fact that there are
> > two different fingering vectors for a 4:5 does not contradict
> > the fact that there is still transpositional invariance;
>
> Sure.
>
> > I can't see any point in having another generalized keyboard
> > for 22, because with a different keyboard you would also have
> > to learn new fingering patterns.
>
> It seems to me new patterns must be learnt either way.
> But I might agree that a new arrangement of digitals is
> not called for. I have to do more experimentation with it.
>
> > Ah, it just occured to me that if you want the 34 or 58
> > divisions, for example, you'll need a different keyboard,
> > which will also do 22, 46, and 56. I'd call this a pajara
> > keyboard. See:
> > /tuning-math/files/secor/kbds/
>
> Wow, a 22-column layout. Interesting. I think I'd prefer
> something more compact (more along the lines of the 5-column
> layout recently suggested by Kalle) but I'll have to do more
> experimenting.
>
> -Carl
>

🔗Brofessor <kraiggrady@...>

to find all the alternative keyboards for 22 one can take all the possible generators 1/22. 3/22 5/22 7/22 9/22 and find them on the scale tree and look above and one can choose any of these keyboards.
It is not a matter of ONE keyboard as each keyboard will allow different things to be easier or harder depending one what one is doing musically. even in working with just constant structures these different layouts give one new possibilities depending on how one is thinking of the material.

--- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@...> wrote:
>
> This keyboard is a 7/12 keyboard on the scale tree which is expanded from the 4/7 keyboard directly above it.
>
> --- In tuning@yahoogroups.com, "Carl Lumma" <carl@> wrote:
> >
> > Hi George,
> >
> > > Only the 3-limit consonances will map alike for 22, 41, and 31
> > > (comparing doubly-positive 22 with typical divisions that are
> > > singly positive and singly negative, respectively)
> >
> > Yes.
> >
> > > But the Bosanquet keyboard is still generalized *insofar* as
> > > the tunings permit the same fingerings, and the octaves in 22
> > > are still exactly lateral.
> >
> > Fingerings for... the chain of 5ths, but not much else.
> >
> > > My point is that Bosanquet's original generalized keyboard
> > > works just fine for 22; you merely have to learn different
> > > locations for primes 5, 7, and 11. The fact that there are
> > > two different fingering vectors for a 4:5 does not contradict
> > > the fact that there is still transpositional invariance;
> >
> > Sure.
> >
> > > I can't see any point in having another generalized keyboard
> > > for 22, because with a different keyboard you would also have
> > > to learn new fingering patterns.
> >
> > It seems to me new patterns must be learnt either way.
> > But I might agree that a new arrangement of digitals is
> > not called for. I have to do more experimentation with it.
> >
> > > Ah, it just occured to me that if you want the 34 or 58
> > > divisions, for example, you'll need a different keyboard,
> > > which will also do 22, 46, and 56. I'd call this a pajara
> > > keyboard. See:
> > > /tuning-math/files/secor/kbds/
> >
> > Wow, a 22-column layout. Interesting. I think I'd prefer
> > something more compact (more along the lines of the 5-column
> > layout recently suggested by Kalle) but I'll have to do more
> > experimenting.
> >
> > -Carl
> >
>

🔗Carl Lumma <carl@...>

George wrote:

> > http://gfax.ch/files/Literature/music/Bosanquet--
> > On%20the%20Hindoo%20Division%20of%20the%20Octave.pdf
>
> Thanks for locating that. This is the same keyboard geometry
> as the pajara keyboard diagrams I referred to in my previous
> message:
> /tuning/topicId_94391.html#94614

Not sure what you mean by geometry, but your keyboard has less
vertical offset between columns, which means it's tilted to
make the octaves lateral, which means the octave span is
effectively 'wider'. The benefit is you get stuff like //c
closer than Bosanquet does.

> No, it doesn't. Chains of fifths, not Bosanquet-order, is
> what determines the suitability of the keyboard.

Interesting observation. I'll ponder it.

> No one keyboard will do it all, but Bosanquet's original
> generalized keyboard does more than any other.

How do you figure?

-Carl

🔗Carl Lumma <carl@...>

🔗gdsecor <gdsecor@...>

--- In tuning@yahoogroups.com, "Carl Lumma" <carl@...> wrote:
>
> --- In tuning@yahoogroups.com, "Brofessor" <kraiggrady@> wrote:
> >
> > is it possible to move a copy of the file here for those not
> > members of tuning math?
>
> My first reaction was surprise that nonmembers can't access
> the files there. So I went over to change it so nonmembers
> *can* access them, but Yahoo apparently no longer allows
> this option.
>
> But membership there is open, so I suggest you sign up with
> the "no e-mail" or "web only" setting if you don't want to
> read the list. That will give you access to the files section,
> and there are many files there that may be of interest now
> or in the future.
>
> best,
>
> -Carl

I put most of my files there, because there was hardly any space left on this list at the time I uploaded them. I don't want to upload any duplicates to this list, because it is better to have that space available to others.

Sorry for the inconvenience, but as Carl pointed out, joining tuning-math will give you immediate access.

--George

🔗Brofessor <kraiggrady@...>

🔗Graham Breed <gbreed@...>

"gdsecor" <gdsecor@...> wrote:

> I'll have to take your word for it, because I haven't yet
> read the paper. I only had time to look at the keyboard
> diagram, which instantly showed me that it's essentially
> a pajara keyboard. The vertically aligned keys in my
> diagrams are different from Bosanquet's; mine differ by 1
> degree of 34 and 56 and are duplicates in 22.

He describes it as a 5-limit tuning. Pajara is one
extension to the 7-limit, but not the only one of interest.

> The original generalized keyboard will work with
> second-order tunings that have all the tones in a single
> chain of 5ths, in other words, about 1/2 of them. Thus
> 22 and 46 will work, but 34 and 58 won't. This keyboard
> will also work with 56 (probably the optimal division for
> pajara), which is third-order.

Sure, and it won't be generalized across 22 and 56 as a
pajara.

> I don't think he got it exactly right. The suitability
> of the keyboard depends on the number of chains of
> fifths, not on the order of the division (as Bosanquet
> defined it).

He didn't have a full theory of regular temperaments. He
was one number short in his definition -- you need three
for a rank 5 class. He only seems to have been thinking of
fifth/octave generation. But that's reasonable as
meantone, schismatic, and diaschismic were the only
systems he was looking at. A full theory of regular
temperaments isn't easy to find, and if it was obvious to
you, you shouldn't minimize the difficulty it caused for
the rest of us ;-)

> Furthermore, on the first-order keyboard the positive
> first-order divisions don't have primes 5, 7, and 11
> mapped the same way as the negative first-order divisions
> (yet this doesn't disqualify the keyboard from being
> generalized). It's all a matter of "insofar as the
> tunings allow it." If you wanted 31 and 41 to have all
> the primes mapped the same way, you'd need a decimal
> (Miracle) keyboard, which would also cover 72. (The
> decimal keyboard will also accommodate 19 and 53, but the
> mapping of most of the primes is quite different.)

He made the distinction between positive and negative
systems. He knew the 5 was mapped differently but the same
keyboard would work. I don't know how far he went with 7.

Diaschismic tunings already fail to generalize if you
ignore the primes. You can't vary the tuning continuously
and correctly from one to another. There is a distinct
class of double-positive scales.

> > Of course it works with a chain of fifths, because the
> > keyboard is based on a chain of fifths. It doesn't work
> > with tunings that have two distinct chains of fifths.
> > It doesn't work at all for 34, does it?
>
> No, it doesn't. Chains of fifths, not Bosanquet-order,
> is what determines the suitability of the keyboard.

Bosanquet's keyboards (two, or three mappings) all assume
chains of fifths. They won't work with Miracle or
Kleismatic. Bosanquet-order assumes a generator of a fifth.

> No one keyboard will do it all, but Bosanquet's original
> generalized keyboard does more than any other.

I don't agree. Bosanquet's first keyboard does two systems
-- meantone and schismatic -- which happen to be very
important. It doesn't do anything else as well. You can
also use a lattice-based layout, as I said in another
message. That can generalize better but comes with other
disadvantages.

Graham

🔗John Moriarty <JlMoriart@...>

How is the Bosanquet keyboard defined? It seems similar to Fokker, and besides symmetries and slants they seem to have other, relevant differences.

I can't find a good picture of either layout with standard note labeling or generator/period coordinates, like here for the wicki layout:
http://oi53.tinypic.com/2v2z98k.jpg
A similar one for fokker and bosanquet would be very helpful.

John

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> "gdsecor" <gdsecor@...> wrote:
>
> > I'll have to take your word for it, because I haven't yet
> > read the paper. I only had time to look at the keyboard
> > diagram, which instantly showed me that it's essentially
> > a pajara keyboard. The vertically aligned keys in my
> > diagrams are different from Bosanquet's; mine differ by 1
> > degree of 34 and 56 and are duplicates in 22.
>
> He describes it as a 5-limit tuning. Pajara is one
> extension to the 7-limit, but not the only one of interest.
>
> > The original generalized keyboard will work with
> > second-order tunings that have all the tones in a single
> > chain of 5ths, in other words, about 1/2 of them. Thus
> > 22 and 46 will work, but 34 and 58 won't. This keyboard
> > will also work with 56 (probably the optimal division for
> > pajara), which is third-order.
>
> Sure, and it won't be generalized across 22 and 56 as a
> pajara.
>
> > I don't think he got it exactly right. The suitability
> > of the keyboard depends on the number of chains of
> > fifths, not on the order of the division (as Bosanquet
> > defined it).
>
> He didn't have a full theory of regular temperaments. He
> was one number short in his definition -- you need three
> for a rank 5 class. He only seems to have been thinking of
> fifth/octave generation. But that's reasonable as
> meantone, schismatic, and diaschismic were the only
> systems he was looking at. A full theory of regular
> temperaments isn't easy to find, and if it was obvious to
> you, you shouldn't minimize the difficulty it caused for
> the rest of us ;-)
>
> > Furthermore, on the first-order keyboard the positive
> > first-order divisions don't have primes 5, 7, and 11
> > mapped the same way as the negative first-order divisions
> > (yet this doesn't disqualify the keyboard from being
> > generalized). It's all a matter of "insofar as the
> > tunings allow it." If you wanted 31 and 41 to have all
> > the primes mapped the same way, you'd need a decimal
> > (Miracle) keyboard, which would also cover 72. (The
> > decimal keyboard will also accommodate 19 and 53, but the
> > mapping of most of the primes is quite different.)
>
> He made the distinction between positive and negative
> systems. He knew the 5 was mapped differently but the same
> keyboard would work. I don't know how far he went with 7.
>
> Diaschismic tunings already fail to generalize if you
> ignore the primes. You can't vary the tuning continuously
> and correctly from one to another. There is a distinct
> class of double-positive scales.
>
> > > Of course it works with a chain of fifths, because the
> > > keyboard is based on a chain of fifths. It doesn't work
> > > with tunings that have two distinct chains of fifths.
> > > It doesn't work at all for 34, does it?
> >
> > No, it doesn't. Chains of fifths, not Bosanquet-order,
> > is what determines the suitability of the keyboard.
>
> Bosanquet's keyboards (two, or three mappings) all assume
> chains of fifths. They won't work with Miracle or
> Kleismatic. Bosanquet-order assumes a generator of a fifth.
>
> > No one keyboard will do it all, but Bosanquet's original
> > generalized keyboard does more than any other.
>
> I don't agree. Bosanquet's first keyboard does two systems
> -- meantone and schismatic -- which happen to be very
> important. It doesn't do anything else as well. You can
> also use a lattice-based layout, as I said in another
> message. That can generalize better but comes with other
> disadvantages.
>
>
> Graham
>

🔗Graham Breed <gbreed@...>

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> "John Moriarty" <JlMoriart@...> wrote:
> > How is the Bosanquet keyboard defined? It seems similar
> > to Fokker, and besides symmetries and slants they seem to
> > have other, relevant differences.
>
> Yes, that's about it. Bosanquet was first, so it's a
> Bosanquet layout, although it's essentially an accordion
> layout:

The closest thing I know of in accordions is the
B-system free bass

http://upload.wikimedia.org/wikipedia/commons/2/25/Melodiebass.jpg

which is basically a post-Bosanquet development, though
Wikipedia suggests the earliest versions may go back to
the 1850s.

http://en.wikipedia.org/wiki/Schrammel_accordion

Bosanquet was certainly right in there.

-Carl

🔗Graham Breed <gbreed@...>

🔗John Moriarty <JlMoriart@...>

> > How is the Bosanquet keyboard defined? It seems similar
> > to Fokker, and besides symmetries and slants they seem to
> > have other, relevant differences.
>
> Yes, that's about it. Bosanquet was first, so it's a
> Bosanquet layout, although it's essentially an accordion
> layout:

Ok, cool. So then this file is relevant to the discussion:

http://f1.grp.yahoofs.com/v1/AJ_mTAxsBCyLqjiu8MDF_piIw3QApwtK8dTz9j7nnsrSf8HAKiJkBPSy7NAodgY5ODNxRIofocAKkywQwkU-cOWoiW_u/JohnMoriarty/SwatheThickness.png

It graphs swathe thickness of different keyboard layouts in relation to generator size.
There was a comment (though I can't seem to find it) that had the gist of

> "No keyboard can do everything, but Bosanquet did the most the best"

or something like that.

This file shows that the Bosanquet (Fokker) layout is extremely efficient in placing the octave reduced instances of intervals generated by a continuous chain of fifths close together (which I'm pretty is what basically defines swath thickness). Generators not around the optimal meantone range, however, are not supported near as well. That is, one would need a much larger Fokker keyboard to host scales of the same cardinality with a 400 cent generator compared to a 700 cent generator.

Conversely, the mirror of L2 has a max in the 400 cent range and is optimal for the magic temperament, but would require a far larger keyboard to be useful for meantone.

The Wicki layout, on the other hand, has a reasonable swathe thickness over all generators other than the first and last 90 cents. This makes it a pretty good compromise for all temperaments, providing a harmonic/melodic gamut comparable over most all popular temperaments. If one were to play a very harmonically adventurous piece, however, one would fall of the edge of the keyboard much sooner than if one were to use an optimal keyboard for the current temperament.

John

🔗Carl Lumma <carl@...>

Hi John,

> Ok, cool. So then this file is relevant to the discussion:

Ya gotta use this URL

/tuning/files/JohnMoriarty/SwatheThickness.png

What are the units of T? And weren't you just asking what
the Fokker layout was -- how'd you plot it if you didn't know?

> It graphs swathe thickness of different keyboard layouts in
> relation to generator size. There was a comment (though I
> can't seem to find it) that had the gist of

This one perhaps
/tuning/topicId_94031.html#94301

or this one?
/tuning/topicId_94391.html#94624

> This file shows that the Bosanquet (Fokker) layout is
> extremely efficient in placing the octave reduced instances of
> intervals generated by a continuous chain of fifths close
> together (which I'm pretty is what basically defines swath
> thickness).

I guess that depends how you define swath thickness.
How do you define it?

> The Wicki layout, on the other hand, has a reasonable swathe
> thickness over all generators other than the first and
> last 90 cents. This makes it a pretty good compromise for all
> temperaments, providing a harmonic/melodic gamut comparable
> over most all popular temperaments.

I believe Wicki is what Graham would call a "lattice-based"
layout. You give up pitch-height ordering for easy reach
of consonances. It makes a lot of sense to me and it's one
thing I plan to investigate with my AXiS.

-Carl

🔗John Moriarty <JlMoriart@...>

> What are the units of T? And weren't you just asking what
> the Fokker layout was -- how'd you plot it if you didn't know?

This isn't my work, I credited in the files section to Milne, Sethares, and Plamondon but not in the image. Sorry about that. I did already have a pretty good idea about Fokker though! It was whether Bosanquet was the same that I was after.
The units of T? Well... I'm trying to take a look at the vector math, but exactly the geometry of why they define T the way they do (mathematically) I'm not sure of. T is defined as the thickness of the "swathe" on a keyboard layout, the swathe being the chunk of the keyboard composed solely of alpha (octave) reduced members of the beta (generator) chain. So maybe the units are a distance in relation to the original definition of the basis vectors? Anyway, the PDF is here, the derivation of T starting on page 7:
http://www.thummer.com/ThumTone/Tuning_Invariant_Layouts_Last_Draft.pdf

The larger the swathe thickness, the better the modulatory recourses and the bigger the largest MOS scale that can be hosted. This is simultaneously sacrificing range however, and the thicker the T-max the greater the slopes around it so the smaller area over which is is useful. A layout with a low swath thickness at its T-max will have a good range (in pitch) and will also support a larger range of generators.

John

🔗Carl Lumma <carl@...>

--- In tuning@yahoogroups.com, "John Moriarty" <JlMoriart@...> wrote:
>
> > What are the units of T? And weren't you just asking what
> > the Fokker layout was -- how'd you plot it if you didn't know?
>
> This isn't my work, I credited in the files section to Milne,
> Sethares, and Plamondon but not in the image.

Ah, sorry I missed that.

> The units of T? Well... I'm trying to take a look at the vector
> math, but exactly the geometry of why they define T the way
> they do (mathematically) I'm not sure of. T is defined as the
> thickness of the "swathe" on a keyboard layout, the swathe being
> the chunk of the keyboard composed solely of alpha (octave)
> reduced members of the beta (generator) chain. So maybe the
> units are a distance in relation to the original definition of
> the basis vectors?

If you have the X_System paper, it's described very simply
in section 6.2. It's what I've been calling the 'number
of columns'. That is, the keyboard distance to the octave.

> The larger the swathe thickness, the better the modulatory
> recourses and the bigger the largest MOS scale that can be
> hosted. This is simultaneously sacrificing range however,

I think of it as, thicker swaths make it easier to reach
smaller intervals and harder to reach larger ones.

-Carl

🔗gdsecor <gdsecor@...>

Sorry, but I've only been able to check in here every few days for a little while, so I haven't been able to reply quickly -- but better late than never, I hope.

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
> "gdsecor" <gdsecor@...> wrote:
>
> > No one keyboard will do it all, but Bosanquet's original
> > generalized keyboard does more than any other.
>
> I don't agree. Bosanquet's first keyboard does two systems
> -- meantone and schismatic -- which happen to be very
> important. It doesn't do anything else as well. You can
> also use a lattice-based layout, as I said in another
> message. That can generalize better but comes with other
> disadvantages.

I don't agree. Bosanquet's first keyboard does one particular (and very important) pajara division, 22, better than schismatic -- better in the sense that 22 is much easier to play than any schismatic tuning. I make this claim from my many years of experience of playing each of these on an actual generalized keyboard. The reason for this is quite simple: consider, for example, a 4:5:6:7:9-approximated chord. In schismatic the total range from the root tone is 16 generators (positions -14 to +2 in the chain of fifths), whereas in 22 it's only 9 generators (positions 0 to +9). (The best meantone division, 31, spans 10 generators, 0 to +10, but overall it's somewhat easier than 22, due to its simpler 5-limit triad range of 4 generators.) The tunings (or more accurately, the divisions) that are the most difficult to play are the ones that require the greatest amount of reaching in the y-direction (46 being a good example).

A keyboard that will accommodate 12, 17, 19, 22, 29, 31, and 41 as well as (if not better than) anything else is, in my book, a tough act to beat.

IMO the disadvantages of a lattice-based layout amount to not doing anything other than 12 particularly well.

--George

🔗Graham Breed <gbreed@...>

On 21 November 2010 07:21, gdsecor <gdsecor@...> wrote:
> Sorry, but I've only been able to check in here every few days for a little while, so I haven't been able to reply quickly -- but better late than never, I hope.

I've had time to read more of Bosanquet, anyway. He explicitly gives
the 7-limit mappings. His negative mapping is for the usual septimal
meantone (he only uses "meantone" for the 1/4-comma tuning). His
positive mapping is for the simple schismatic we sometimes call
"Garibaldi". He talks about 7-limit intervals but nothing beyond
that.

> I don't agree. Bosanquet's first keyboard does one
> particular (and very important) pajara division, 22,
> better than schismatic -- better in the sense that 22
> is much easier to play than any schismatic tuning.
> I make this claim from my many years of experience
> of playing each of these on an actual generalized
> keyboard. The reason for this is quite simple:
> consider, for example, a 4:5:6:7:9-approximated
> chord. In schismatic the total range from the root
> tone is 16 generators (positions -14 to +2 in the chain
> of fifths), whereas in 22 it's only 9 generators
> (positions 0 to +9). (The best meantone division, 31,
> spans 10 generators, 0 to +10, but overall it's
> somewhat easier than 22, due to its simpler 5-limit
> triad range of 4 generators.) The tunings (or more
> accurately, the divisions) that are the most difficult to
> play are the ones that require the greatest amount of
> reaching in the y-direction (46 being a good example).

This is Superpyth, isn't it? So fine, the Bosanquet layout does
Superpyth as well, and maybe some other minor temperaments. (I don't
know if Mavila would work.) It doesn't do more than any other
MOS-based layout. It's special because the things it does are
important.

Superpyth is simpler than Schismatic, but much less accurate. I don't
know why you'd bother with it if you didn't already have a keyboard
that happened to be designed around fifths and octaves. If you want
simple 9-limit harmony, and you're happy with the approximation,
counting generators suggests you'd be better off with a true Pajara
keyboard. The 9-limit is 8 generators/octave in Pajara compared with
11 in Superpyth (not 9, either you miscounted the 7:5, or there's
something strange going on). If there's some specific feature of the
mapping that makes Superpyth better, fine, but it still isn't Pajara.

> A keyboard that will accommodate 12, 17, 19, 22, 29,
> 31, and 41 as well as (if not better than) anything else
> is, in my book, a tough act to beat.

Yes, we know Bosanquet's good. If you want to discuss an alternative,
thirds are the place to look. Important temperaments will have
thirds. Hanson and Mohajira will already get you a long way, provided
you're happy with learning more than one fingering. If it can cover
both Orwell and Magic, it'd be very handy in the 11-limit. One equal
temperament you lose is 12, but you can still get it as every other
step of 24.

> IMO the disadvantages of a lattice-based layout
> amount to not doing anything other than 12
> particularly well.

There are plenty to look at. I thought the 5-limit Wicki-style 19
note mapping was promising but I don't have experience of it. I did
tune up a 7-limit neutral thirds mapping (which as been called a
"Breed lattice") on my Ztar. It has one generator as the approximate
11:9 (or extended 7-limit neutral third if you don't want the
11-limit) and the other the approximate 10:7. I still preferred
melodic mappings. And the more primes you squeeze in the less
generalization you get, anyway.

Graham

🔗genewardsmith <genewardsmith@...>

--- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:

> There are plenty to look at. I thought the 5-limit Wicki-style 19
> note mapping was promising but I don't have experience of it. I did
> tune up a 7-limit neutral thirds mapping (which as been called a
> "Breed lattice") on my Ztar. It has one generator as the approximate
> 11:9 (or extended 7-limit neutral third if you don't want the
> 11-limit) and the other the approximate 10:7. I still preferred
> melodic mappings. And the more primes you squeeze in the less
> generalization you get, anyway.

5-limit and Breed planar lattices are for pitch classes, so I'm wondering if you are talking about using a particular scale, such as some 19-note scale for the 5-limit. One great thing about rank two temperaments are keyboards, I suppose. Rank one temperaments can be treated as if they are rank two, but how do you "squeeze in" more generators?

🔗gdsecor <gdsecor@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@> wrote:
>
> > There are plenty to look at. I thought the 5-limit Wicki-style 19
> > note mapping was promising but I don't have experience of it. I did
> > tune up a 7-limit neutral thirds mapping (which as been called a
> > "Breed lattice") on my Ztar. It has one generator as the approximate
> > 11:9 (or extended 7-limit neutral third if you don't want the
> > 11-limit) and the other the approximate 10:7. I still preferred
> > melodic mappings. And the more primes you squeeze in the less
> > generalization you get, anyway.
>
> 5-limit and Breed planar lattices are for pitch classes, so I'm wondering if you are talking about using a particular scale, such as some 19-note scale for the 5-limit. One great thing about rank two temperaments are keyboards, I suppose. Rank one temperaments can be treated as if they are rank two, but how do you "squeeze in" more generators?

Aaron Hunt's Tonal Plexus keyboard can easily handle rank three tunings, as long as you don't require the third dimension to extend any farther than five tones. For example, you can use an octave, the tempered fifth of 31-equal, and a single degree of 217-equal (~1/4-comma) as generators to play 31-equal with flexible intonation, using a technique similar to adaptive-JI.

You can also map divisions with multiple chains of fifths (e.g., 34, 58, 87, 159).

I worked out all of this (and more) on a spreadsheet, Plexus255a.xls, that you can download from here:
/tuning-math/files/secor/kbds/
Enter the desired parameters in the three colored cells at the upper left. Use the table at the right to find the appropriate parameters for various EDO's. The numbers that appear in each cell of the keyboard diagram indicate degrees of the EDO entered in cell A2.

The JND parameter corresponds to number of degrees of the octave division for the 3rd-dimension generator. I used 0.3 and 0.7 to specify ~1/3 and ~2/3, respectively, and the calculation for each keyboard cell is rounded to a whole number of degrees. Observe that many of these tunings will have some vertically adjacent keys tuned alike.

--George

🔗genewardsmith <genewardsmith@...>

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

> Aaron Hunt's Tonal Plexus keyboard can easily handle rank three tunings, as long as you don't require the third dimension to extend any farther than five tones. For example, you can use an octave, the tempered fifth of 31-equal, and a single degree of 217-equal (~1/4-comma) as generators to play 31-equal with flexible intonation, using a technique similar to adaptive-JI.

That third interval is pretty small.

It occurs to me that one way to treat rank three temperaments is with wolves. For an example, myna tempers out 2401/2400, and you can make a keyboard layout for it by using myna-tempered 10/9 and 8/7 as generators. If you do this, (8/7)^11 (9/10)^7 gives the octave. Instead of using myna tempering, you could use breed tempering. For instance, the minimax tuning has the 5-limit just, and 7 flat by (2401/2400)^(1/4), so the generators could be a just 10/9 and an 8/7 sharp by 0.18 cents. You can go along happily until to reach 2, where the step sizes change--they are wolf steps. This is probably not a great example, since the method is likely to work best if the size adjustment is a little less drastic.

🔗gdsecor <gdsecor@...>

--- In tuning@yahoogroups.com, "genewardsmith" <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, "gdsecor" <gdsecor@> wrote:
>
> > Aaron Hunt's Tonal Plexus keyboard can easily handle rank three tunings, as long as you don't require the third dimension to extend any farther than five tones. For example, you can use an octave, the tempered fifth of 31-equal, and a single degree of 217-equal (~1/4-comma) as generators to play 31-equal with flexible intonation, using a technique similar to adaptive-JI.
>
> That third interval is pretty small.

I don't think that it would necessarily have to be that small, but using a much large interval between vertically adjacent keys could be rather confusing. OTOH, there's nothing forcing you to map the 3rd generator on adjacent keys (inasmuch as the other two are not on adjacent keys). For example, take the octave and fifth of 17-equal as generators and 13 degrees of 68-equal as your third generator (enter parameters 68,1,0 on the spreadsheet). The physical vector for the third generator would then be 2 positions right and 2 positions up. (There's also nothing restricting you to exact degrees of an EDO; these are merely examples.)

--George

🔗Graham Breed <gbreed@...>

On 22 November 2010 23:22, genewardsmith <genewardsmith@...> wrote:
>
> --- In tuning@yahoogroups.com, Graham Breed <gbreed@...> wrote:
>
>> There are plenty to look at. I thought the 5-limit Wicki-style 19
>> note mapping was promising but I don't have experience of it. I did
>> tune up a 7-limit neutral thirds mapping (which as been called a
>> "Breed lattice") on my Ztar. It has one generator as the approximate
>> 11:9 (or extended 7-limit neutral third if you don't want the
>> 11-limit) and the other the approximate 10:7. I still preferred
>> melodic mappings. And the more primes you squeeze in the less
>> generalization you get, anyway.
>
> 5-limit and Breed planar lattices are for pitch classes, so I'm
> wondering if you are talking about using a particular scale,
> such as some 19-note scale for the 5-limit. One great thing about
> rank two temperaments are keyboards, I suppose. Rank one
> temperaments can be treated as if they are rank two, but how
> do you "squeeze in" more generators?

I can foresee applications for octave-equivalent keyboards (keyboards
with pitch classes) when you use them with computers. You could have
some kind of auto accompaniment program choosing the inversions. Or
have the chords from one instrument control the tuning of other
instruments. Or you can take the lattice at face value, give a pitch
to each key (as I describe for "Breed", or based on root position
triads for the usual 5-limit lattice) and give up having a choice of
inversion. This might make sense for a split keyboard, where the left
hand plays chords, and the right hand uses a completely different
mapping.

Generally we'd be talking about identifying an approximate octave, and
setting it to be exact. This means adding a unison vector (explicitly
an "octave vector") and giving a rank 2 temperament. So at that point
the keyboard's no more generalized than any other rank 2 layout. But
it's still biased toward chords with one particular inversion, and
chord sequences that work in the planar tuning with that inversion can
generalize to different tunings.

In general with "Breed" you'd need both hands to get specific chord
inversions. Which isn't so bad, because I don't know of a keyboard
that does give you arbitrary inversions of 11-limit chords exceeding
an octave with one hand.

Generalized keyboards are rank 2 beasts. The keyboard is always a two
dimensional surface. There are basic topological reasons why you
can't escape that. There are ways of getting higher rank tunings, as
George says, but you lose transposability.

My "Wicki-style" suggestion was about defining a rank 1 object --
essentially a periodicity block. You then have a keyboard for the
corresponding equal temperament, but also for rank 2 temperament
classes that include it. If it's based on a lattice, any chord in the
generalized root position will either be playable with the same
fingering, or not be playable at all because it lies outside the MOS
you've tuned to. For other inversions, you can make use of two
different "octave vectors" so that you have a choice of fingering.
And whatever fingering you choose will either work or not work
depending on whether the chord belongs to the MOS you've tuned to.

We all seem to agree that the Wicki layout itself its good for 12 note
scales. My other suggestion was to extend it to 19 notes, and this is
what George disagreed with. With 19 notes, you can keep the
black/white coloring to help orientation, because 19 is a meantone.
You can also use conventional notation. But you can also generalize
the keyboard to Magic and Hanson without having to re-learn
fingerings. Any chords or tunes that happen to work with both scales
will also work with the same fingering.

Because of the lattice geometry, you still get a (vertical) line of
approximate 25:24 intervals, which are always steps in the MOS. A 31
note layout, as well as being less manageable, might not be so useful
melodically because the approximate 25:24 isn't such a useful
interval. I don't know how a 22 note keyboard would work out. It
would, at least, mean you lose the familiarity of black/white keys.

Of course, if you want more notes, you can relax one of the octave
vectors to get a rank 2 layout. But you then lose generality between
different temperament classes and if you want to share a keyboard with
different mappings the key colors would be confusing.

You can't squeeze in more generators. You can only squeeze in more
primes with the current generators, which means implying unison
vectors. So the keyboard stops being generalized across different
temperament classes. The Breed lattice with 11:9 neutral thirds is
also pretty much restricted to Miracle. As such it's not more
generalized than any other Miracle layout.

Graham